2. variables
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5. variables
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8. variables
The simplest way to understand a variable is as any characteristic or attribute that can experience change or vary over time or context – hence the name “variable”. For example, the dosage of a particular medicine could be classified as a variable, as the amount can vary (i.e., a higher dose or a lower dose). Similarly, gender, age or ethnicity could be considered demographic variables, because each person varies in these respects.
Within research, especially scientific research, variables form the foundation of studies, as researchers are often interested in how one variable impacts another, and the relationships between different variables. For example:
As you can see, variables are often used to explain relationships between different elements and phenomena. In scientific studies, especially experimental studies, the objective is often to understand the causal relationships between variables. In other words, the role of cause and effect between variables. This is achieved by manipulating certain variables while controlling others – and then observing the outcome. But, we’ll get into that a little later…
Variables can be a little intimidating for new researchers because there are a wide variety of variables, and oftentimes, there are multiple labels for the same thing. To lay a firm foundation, we’ll first look at the three main types of variables, namely:
Simply put, the independent variable is the “ cause ” in the relationship between two (or more) variables. In other words, when the independent variable changes, it has an impact on another variable.
For example:
It’s useful to know that independent variables can go by a few different names, including, explanatory variables (because they explain an event or outcome) and predictor variables (because they predict the value of another variable). Terminology aside though, the most important takeaway is that independent variables are assumed to be the “cause” in any cause-effect relationship. As you can imagine, these types of variables are of major interest to researchers, as many studies seek to understand the causal factors behind a phenomenon.
While the independent variable is the “ cause ”, the dependent variable is the “ effect ” – or rather, the affected variable . In other words, the dependent variable is the variable that is assumed to change as a result of a change in the independent variable.
Keeping with the previous example, let’s look at some dependent variables in action:
In scientific studies, researchers will typically pay very close attention to the dependent variable (or variables), carefully measuring any changes in response to hypothesised independent variables. This can be tricky in practice, as it’s not always easy to reliably measure specific phenomena or outcomes – or to be certain that the actual cause of the change is in fact the independent variable.
As the adage goes, correlation is not causation . In other words, just because two variables have a relationship doesn’t mean that it’s a causal relationship – they may just happen to vary together. For example, you could find a correlation between the number of people who own a certain brand of car and the number of people who have a certain type of job. Just because the number of people who own that brand of car and the number of people who have that type of job is correlated, it doesn’t mean that owning that brand of car causes someone to have that type of job or vice versa. The correlation could, for example, be caused by another factor such as income level or age group, which would affect both car ownership and job type.
To confidently establish a causal relationship between an independent variable and a dependent variable (i.e., X causes Y), you’ll typically need an experimental design , where you have complete control over the environmen t and the variables of interest. But even so, this doesn’t always translate into the “real world”. Simply put, what happens in the lab sometimes stays in the lab!
As an alternative to pure experimental research, correlational or “ quasi-experimental ” research (where the researcher cannot manipulate or change variables) can be done on a much larger scale more easily, allowing one to understand specific relationships in the real world. These types of studies also assume some causality between independent and dependent variables, but it’s not always clear. So, if you go this route, you need to be cautious in terms of how you describe the impact and causality between variables and be sure to acknowledge any limitations in your own research.
In an experimental design, a control variable (or controlled variable) is a variable that is intentionally held constant to ensure it doesn’t have an influence on any other variables. As a result, this variable remains unchanged throughout the course of the study. In other words, it’s a variable that’s not allowed to vary – tough life 🙂
As we mentioned earlier, one of the major challenges in identifying and measuring causal relationships is that it’s difficult to isolate the impact of variables other than the independent variable. Simply put, there’s always a risk that there are factors beyond the ones you’re specifically looking at that might be impacting the results of your study. So, to minimise the risk of this, researchers will attempt (as best possible) to hold other variables constant . These factors are then considered control variables.
Some examples of variables that you may need to control include:
Which specific variables need to be controlled for will vary tremendously depending on the research project at hand, so there’s no generic list of control variables to consult. As a researcher, you’ll need to think carefully about all the factors that could vary within your research context and then consider how you’ll go about controlling them. A good starting point is to look at previous studies similar to yours and pay close attention to which variables they controlled for.
Of course, you won’t always be able to control every possible variable, and so, in many cases, you’ll just have to acknowledge their potential impact and account for them in the conclusions you draw. Every study has its limitations , so don’t get fixated or discouraged by troublesome variables. Nevertheless, always think carefully about the factors beyond what you’re focusing on – don’t make assumptions!
As we mentioned, independent, dependent and control variables are the most common variables you’ll come across in your research, but they’re certainly not the only ones you need to be aware of. Next, we’ll look at a few “secondary” variables that you need to keep in mind as you design your research.
Let’s jump into it…
A moderating variable is a variable that influences the strength or direction of the relationship between an independent variable and a dependent variable. In other words, moderating variables affect how much (or how little) the IV affects the DV, or whether the IV has a positive or negative relationship with the DV (i.e., moves in the same or opposite direction).
For example, in a study about the effects of sleep deprivation on academic performance, gender could be used as a moderating variable to see if there are any differences in how men and women respond to a lack of sleep. In such a case, one may find that gender has an influence on how much students’ scores suffer when they’re deprived of sleep.
It’s important to note that while moderators can have an influence on outcomes , they don’t necessarily cause them ; rather they modify or “moderate” existing relationships between other variables. This means that it’s possible for two different groups with similar characteristics, but different levels of moderation, to experience very different results from the same experiment or study design.
Mediating variables are often used to explain the relationship between the independent and dependent variable (s). For example, if you were researching the effects of age on job satisfaction, then education level could be considered a mediating variable, as it may explain why older people have higher job satisfaction than younger people – they may have more experience or better qualifications, which lead to greater job satisfaction.
Mediating variables also help researchers understand how different factors interact with each other to influence outcomes. For instance, if you wanted to study the effect of stress on academic performance, then coping strategies might act as a mediating factor by influencing both stress levels and academic performance simultaneously. For example, students who use effective coping strategies might be less stressed but also perform better academically due to their improved mental state.
In addition, mediating variables can provide insight into causal relationships between two variables by helping researchers determine whether changes in one factor directly cause changes in another – or whether there is an indirect relationship between them mediated by some third factor(s). For instance, if you wanted to investigate the impact of parental involvement on student achievement, you would need to consider family dynamics as a potential mediator, since it could influence both parental involvement and student achievement simultaneously.
A confounding variable (also known as a third variable or lurking variable ) is an extraneous factor that can influence the relationship between two variables being studied. Specifically, for a variable to be considered a confounding variable, it needs to meet two criteria:
Some common examples of confounding variables include demographic factors such as gender, ethnicity, socioeconomic status, age, education level, and health status. In addition to these, there are also environmental factors to consider. For example, air pollution could confound the impact of the variables of interest in a study investigating health outcomes.
Naturally, it’s important to identify as many confounding variables as possible when conducting your research, as they can heavily distort the results and lead you to draw incorrect conclusions . So, always think carefully about what factors may have a confounding effect on your variables of interest and try to manage these as best you can.
Latent variables are unobservable factors that can influence the behaviour of individuals and explain certain outcomes within a study. They’re also known as hidden or underlying variables , and what makes them rather tricky is that they can’t be directly observed or measured . Instead, latent variables must be inferred from other observable data points such as responses to surveys or experiments.
For example, in a study of mental health, the variable “resilience” could be considered a latent variable. It can’t be directly measured , but it can be inferred from measures of mental health symptoms, stress, and coping mechanisms. The same applies to a lot of concepts we encounter every day – for example:
One way in which we overcome the challenge of measuring the immeasurable is latent variable models (LVMs). An LVM is a type of statistical model that describes a relationship between observed variables and one or more unobserved (latent) variables. These models allow researchers to uncover patterns in their data which may not have been visible before, thanks to their complexity and interrelatedness with other variables. Those patterns can then inform hypotheses about cause-and-effect relationships among those same variables which were previously unknown prior to running the LVM. Powerful stuff, we say!
In the world of scientific research, there’s no shortage of variable types, some of which have multiple names and some of which overlap with each other. In this post, we’ve covered some of the popular ones, but remember that this is not an exhaustive list .
To recap, we’ve explored:
If you’re still feeling a bit lost and need a helping hand with your research project, check out our 1-on-1 coaching service , where we guide you through each step of the research journey. Also, be sure to check out our free dissertation writing course and our collection of free, fully-editable chapter templates .
This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...
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Chi-squared tests in r.
Chi-Squared Table
All Modules
Here we extend that application of the chi-square test to the case with two or more independent comparison groups. Specifically, the outcome of interest is discrete with two or more responses and the responses can be ordered or unordered (i.e., the outcome can be dichotomous, ordinal or categorical). We now consider the situation where there are two or more independent comparison groups and the goal of the analysis is to compare the distribution of responses to the discrete outcome variable among several independent comparison groups.
The test is called the χ 2 test of independence and the null hypothesis is that there is no difference in the distribution of responses to the outcome across comparison groups. This is often stated as follows: The outcome variable and the grouping variable (e.g., the comparison treatments or comparison groups) are independent (hence the name of the test). Independence here implies homogeneity in the distribution of the outcome among comparison groups.
The null hypothesis in the χ 2 test of independence is often stated in words as: H 0 : The distribution of the outcome is independent of the groups. The alternative or research hypothesis is that there is a difference in the distribution of responses to the outcome variable among the comparison groups (i.e., that the distribution of responses "depends" on the group). In order to test the hypothesis, we measure the discrete outcome variable in each participant in each comparison group. The data of interest are the observed frequencies (or number of participants in each response category in each group). The formula for the test statistic for the χ 2 test of independence is given below.
Test Statistic for Testing H 0 : Distribution of outcome is independent of groups
and we find the critical value in a table of probabilities for the chi-square distribution with df=(r-1)*(c-1).
Here O = observed frequency, E=expected frequency in each of the response categories in each group, r = the number of rows in the two-way table and c = the number of columns in the two-way table. r and c correspond to the number of comparison groups and the number of response options in the outcome (see below for more details). The observed frequencies are the sample data and the expected frequencies are computed as described below. The test statistic is appropriate for large samples, defined as expected frequencies of at least 5 in each of the response categories in each group.
The data for the χ 2 test of independence are organized in a two-way table. The outcome and grouping variable are shown in the rows and columns of the table. The sample table below illustrates the data layout. The table entries (blank below) are the numbers of participants in each group responding to each response category of the outcome variable.
Table - Possible outcomes are are listed in the columns; The groups being compared are listed in rows.
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In the table above, the grouping variable is shown in the rows of the table; r denotes the number of independent groups. The outcome variable is shown in the columns of the table; c denotes the number of response options in the outcome variable. Each combination of a row (group) and column (response) is called a cell of the table. The table has r*c cells and is sometimes called an r x c ("r by c") table. For example, if there are 4 groups and 5 categories in the outcome variable, the data are organized in a 4 X 5 table. The row and column totals are shown along the right-hand margin and the bottom of the table, respectively. The total sample size, N, can be computed by summing the row totals or the column totals. Similar to ANOVA, N does not refer to a population size here but rather to the total sample size in the analysis. The sample data can be organized into a table like the above. The numbers of participants within each group who select each response option are shown in the cells of the table and these are the observed frequencies used in the test statistic.
The test statistic for the χ 2 test of independence involves comparing observed (sample data) and expected frequencies in each cell of the table. The expected frequencies are computed assuming that the null hypothesis is true. The null hypothesis states that the two variables (the grouping variable and the outcome) are independent. The definition of independence is as follows:
Two events, A and B, are independent if P(A|B) = P(A), or equivalently, if P(A and B) = P(A) P(B).
The second statement indicates that if two events, A and B, are independent then the probability of their intersection can be computed by multiplying the probability of each individual event. To conduct the χ 2 test of independence, we need to compute expected frequencies in each cell of the table. Expected frequencies are computed by assuming that the grouping variable and outcome are independent (i.e., under the null hypothesis). Thus, if the null hypothesis is true, using the definition of independence:
P(Group 1 and Response Option 1) = P(Group 1) P(Response Option 1).
The above states that the probability that an individual is in Group 1 and their outcome is Response Option 1 is computed by multiplying the probability that person is in Group 1 by the probability that a person is in Response Option 1. To conduct the χ 2 test of independence, we need expected frequencies and not expected probabilities . To convert the above probability to a frequency, we multiply by N. Consider the following small example.
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| 10 | 8 | 7 | 25 |
| 22 | 15 | 13 | 50 |
| 30 | 28 | 17 | 75 |
| 62 | 51 | 37 | 150 |
The data shown above are measured in a sample of size N=150. The frequencies in the cells of the table are the observed frequencies. If Group and Response are independent, then we can compute the probability that a person in the sample is in Group 1 and Response category 1 using:
P(Group 1 and Response 1) = P(Group 1) P(Response 1),
P(Group 1 and Response 1) = (25/150) (62/150) = 0.069.
Thus if Group and Response are independent we would expect 6.9% of the sample to be in the top left cell of the table (Group 1 and Response 1). The expected frequency is 150(0.069) = 10.4. We could do the same for Group 2 and Response 1:
P(Group 2 and Response 1) = P(Group 2) P(Response 1),
P(Group 2 and Response 1) = (50/150) (62/150) = 0.138.
The expected frequency in Group 2 and Response 1 is 150(0.138) = 20.7.
Thus, the formula for determining the expected cell frequencies in the χ 2 test of independence is as follows:
Expected Cell Frequency = (Row Total * Column Total)/N.
The above computes the expected frequency in one step rather than computing the expected probability first and then converting to a frequency.
In a prior example we evaluated data from a survey of university graduates which assessed, among other things, how frequently they exercised. The survey was completed by 470 graduates. In the prior example we used the χ 2 goodness-of-fit test to assess whether there was a shift in the distribution of responses to the exercise question following the implementation of a health promotion campaign on campus. We specifically considered one sample (all students) and compared the observed distribution to the distribution of responses the prior year (a historical control). Suppose we now wish to assess whether there is a relationship between exercise on campus and students' living arrangements. As part of the same survey, graduates were asked where they lived their senior year. The response options were dormitory, on-campus apartment, off-campus apartment, and at home (i.e., commuted to and from the university). The data are shown below.
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| 32 | 30 | 28 | 90 |
| 74 | 64 | 42 | 180 |
| 110 | 25 | 15 | 150 |
| 39 | 6 | 5 | 50 |
| 255 | 125 | 90 | 470 |
Based on the data, is there a relationship between exercise and student's living arrangement? Do you think where a person lives affect their exercise status? Here we have four independent comparison groups (living arrangement) and a discrete (ordinal) outcome variable with three response options. We specifically want to test whether living arrangement and exercise are independent. We will run the test using the five-step approach.
H 0 : Living arrangement and exercise are independent
H 1 : H 0 is false. α=0.05
The null and research hypotheses are written in words rather than in symbols. The research hypothesis is that the grouping variable (living arrangement) and the outcome variable (exercise) are dependent or related.
The formula for the test statistic is:
The condition for appropriate use of the above test statistic is that each expected frequency is at least 5. In Step 4 we will compute the expected frequencies and we will ensure that the condition is met.
The decision rule depends on the level of significance and the degrees of freedom, defined as df = (r-1)(c-1), where r and c are the numbers of rows and columns in the two-way data table. The row variable is the living arrangement and there are 4 arrangements considered, thus r=4. The column variable is exercise and 3 responses are considered, thus c=3. For this test, df=(4-1)(3-1)=3(2)=6. Again, with χ 2 tests there are no upper, lower or two-tailed tests. If the null hypothesis is true, the observed and expected frequencies will be close in value and the χ 2 statistic will be close to zero. If the null hypothesis is false, then the χ 2 statistic will be large. The rejection region for the χ 2 test of independence is always in the upper (right-hand) tail of the distribution. For df=6 and a 5% level of significance, the appropriate critical value is 12.59 and the decision rule is as follows: Reject H 0 if c 2 > 12.59.
We now compute the expected frequencies using the formula,
Expected Frequency = (Row Total * Column Total)/N.
The computations can be organized in a two-way table. The top number in each cell of the table is the observed frequency and the bottom number is the expected frequency. The expected frequencies are shown in parentheses.
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| 32 (48.8) | 30 (23.9) | 28 (17.2) | 90 |
| 74 (97.7) | 64 (47.9) | 42 (34.5) | 180 |
| 110 (81.4) | 25 (39.9) | 15 (28.7) | 150 |
| 39 (27.1) | 6 (13.3) | 5 (9.6) | 50 |
| 255 | 125 | 90 | 470 |
Notice that the expected frequencies are taken to one decimal place and that the sums of the observed frequencies are equal to the sums of the expected frequencies in each row and column of the table.
Recall in Step 2 a condition for the appropriate use of the test statistic was that each expected frequency is at least 5. This is true for this sample (the smallest expected frequency is 9.6) and therefore it is appropriate to use the test statistic.
The test statistic is computed as follows:
We reject H 0 because 60.5 > 12.59. We have statistically significant evidence at a =0.05 to show that H 0 is false or that living arrangement and exercise are not independent (i.e., they are dependent or related), p < 0.005.
Again, the χ 2 test of independence is used to test whether the distribution of the outcome variable is similar across the comparison groups. Here we rejected H 0 and concluded that the distribution of exercise is not independent of living arrangement, or that there is a relationship between living arrangement and exercise. The test provides an overall assessment of statistical significance. When the null hypothesis is rejected, it is important to review the sample data to understand the nature of the relationship. Consider again the sample data.
Because there are different numbers of students in each living situation, it makes the comparisons of exercise patterns difficult on the basis of the frequencies alone. The following table displays the percentages of students in each exercise category by living arrangement. The percentages sum to 100% in each row of the table. For comparison purposes, percentages are also shown for the total sample along the bottom row of the table.
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| 36% | 33% | 31% |
| 41% | 36% | 23% |
| 73% | 17% | 10% |
| 78% | 12% | 10% |
| 54% | 27% | 19% |
From the above, it is clear that higher percentages of students living in dormitories and in on-campus apartments reported regular exercise (31% and 23%) as compared to students living in off-campus apartments and at home (10% each).
Test Yourself
( J Gastrointest Surgery, 2012, 16 275-281)', CAPTION, '
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0-4 | 21 | 20 | 16 |
5-6 | 135 | 71 | 35 |
7-10 | 158 | 62 | 35 |
Question: What would be an appropriate statistical test to examine whether there is an association between Surgical Apgar Score and patient outcome? Using 14.13 as the value of the test statistic for these data, carry out the appropriate test at a 5% level of significance. Show all parts of your test.
In the module on hypothesis testing for means and proportions , we discussed hypothesis testing applications with a dichotomous outcome variable and two independent comparison groups. We presented a test using a test statistic Z to test for equality of independent proportions. The chi-square test of independence can also be used with a dichotomous outcome and the results are mathematically equivalent.
In the prior module, we considered the following example. Here we show the equivalence to the chi-square test of independence.
A randomized trial is designed to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery. The trial compares the new pain reliever to the pain reliever currently in use (called the standard of care). A total of 100 patients undergoing joint replacement surgery agreed to participate in the trial. Patients were randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery and were blind to the treatment assignment. Before receiving the assigned treatment, patients were asked to rate their pain on a scale of 0-10 with higher scores indicative of more pain. Each patient was then given the assigned treatment and after 30 minutes was again asked to rate their pain on the same scale. The primary outcome was a reduction in pain of 3 or more scale points (defined by clinicians as a clinically meaningful reduction). The following data were observed in the trial.
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| 50 | 23 | 0.46 |
| 50 | 11 | 0.22 |
We tested whether there was a significant difference in the proportions of patients reporting a meaningful reduction (i.e., a reduction of 3 or more scale points) using a Z statistic, as follows.
H 0 : p 1 = p 2
H 1 : p 1 ≠ p 2 α=0.05
Here the new or experimental pain reliever is group 1 and the standard pain reliever is group 2.
We must first check that the sample size is adequate. Specifically, we need to ensure that we have at least 5 successes and 5 failures in each comparison group or that:
In this example, we have
Therefore, the sample size is adequate, so the following formula can be used:
Reject H 0 if Z < -1.960 or if Z > 1.960.
We now substitute the sample data into the formula for the test statistic identified in Step 2. We first compute the overall proportion of successes:
We now substitute to compute the test statistic.
We now conduct the same test using the chi-square test of independence.
H 0 : Treatment and outcome (meaningful reduction in pain) are independent
H 1 : H 0 is false. α=0.05
The formula for the test statistic is:
For this test, df=(2-1)(2-1)=1. At a 5% level of significance, the appropriate critical value is 3.84 and the decision rule is as follows: Reject H0 if χ 2 > 3.84. (Note that 1.96 2 = 3.84, where 1.96 was the critical value used in the Z test for proportions shown above.)
We now compute the expected frequencies using:
The computations can be organized in a two-way table. The top number in each cell of the table is the observed frequency and the bottom number is the expected frequency. The expected frequencies are shown in parentheses.
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| 23 (17.0) | 27 (33.0) | 50 |
| 11 (17.0) | 39 (33.0) | 50 |
| 34 | 66 | 100 |
A condition for the appropriate use of the test statistic was that each expected frequency is at least 5. This is true for this sample (the smallest expected frequency is 22.0) and therefore it is appropriate to use the test statistic.
(Note that (2.53) 2 = 6.4, where 2.53 was the value of the Z statistic in the test for proportions shown above.)
The video below by Mike Marin demonstrates how to perform chi-squared tests in the R programming language.
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Content ©2016. All Rights Reserved. Date last modified: September 1, 2016. Wayne W. LaMorte, MD, PhD, MPH
Have you ever wondered how scientists make discoveries and how researchers come to understand the world around us? A crucial tool in their kit is the concept of the independent variable, which helps them delve into the mysteries of science and everyday life.
An independent variable is a condition or factor that researchers manipulate to observe its effect on another variable, known as the dependent variable. In simpler terms, it’s like adjusting the dials and watching what happens! By changing the independent variable, scientists can see if and how it causes changes in what they are measuring or observing, helping them make connections and draw conclusions.
In this article, we’ll explore the fascinating world of independent variables, journey through their history, examine theories, and look at a variety of examples from different fields.
Once upon a time, in a world thirsty for understanding, people observed the stars, the seas, and everything in between, seeking to unlock the mysteries of the universe.
The story of the independent variable begins with a quest for knowledge, a journey taken by thinkers and tinkerers who wanted to explain the wonders and strangeness of the world.
The seeds of the idea of independent variables were sown by Sir Francis Galton , an English polymath, in the 19th century. Galton wore many hats—he was a psychologist, anthropologist, meteorologist, and a statistician!
It was his diverse interests that led him to explore the relationships between different factors and their effects. Galton was curious—how did one thing lead to another, and what could be learned from these connections?
As Galton delved into the world of statistical theories , the concept of independent variables started taking shape.
He was interested in understanding how characteristics, like height and intelligence, were passed down through generations.
Galton’s work laid the foundation for later thinkers to refine and expand the concept, turning it into an invaluable tool for scientific research.
After Galton’s pioneering work, the concept of the independent variable continued to evolve and grow. Scientists and researchers from various fields adopted and adapted it, finding new ways to use it to make sense of the world.
They discovered that by manipulating one factor (the independent variable), they could observe changes in another (the dependent variable), leading to groundbreaking insights and discoveries.
Through the years, the independent variable became a cornerstone in experimental design . Researchers in fields like physics, biology, psychology, and sociology used it to test hypotheses, develop theories, and uncover the laws that govern our universe.
The idea that originated from Galton’s curiosity had bloomed into a universal key, unlocking doors to knowledge across disciplines.
Today, the independent variable stands tall as a pillar of scientific research. It helps scientists and researchers ask critical questions, test their ideas, and find answers. Without independent variables, we wouldn’t have many of the advancements and understandings that we take for granted today.
The independent variable plays a starring role in experiments, helping us learn about everything from the smallest particles to the vastness of space. It helps researchers create vaccines, understand social behaviors, explore ecological systems, and even develop new technologies.
In the upcoming sections, we’ll dive deeper into what independent variables are, how they work, and how they’re used in various fields.
Together, we’ll uncover the magic of this scientific concept and see how it continues to shape our understanding of the world around us.
Embarking on the captivating journey of scientific exploration requires us to grasp the essential terms and ideas. It's akin to a treasure hunter mastering the use of a map and compass.
In our adventure through the realm of independent variables, we’ll delve deeper into some fundamental concepts and definitions to help us navigate this exciting world.
In the grand tapestry of research, variables are the gems that researchers seek. They’re elements, characteristics, or behaviors that can shift or vary in different circumstances.
Picture them as the myriad of ingredients in a chef’s kitchen—each variable can be adjusted or modified to create a myriad of dishes, each with a unique flavor!
Understanding variables is essential as they form the core of every scientific experiment and observational study.
Independent Variable The star of our story, the independent variable, is the one that researchers change or control to study its effects. It’s like a chef experimenting with different spices to see how each one alters the taste of the soup. The independent variable is the catalyst, the initial spark that sets the wheels of research in motion.
Dependent Variable The dependent variable is the outcome we observe and measure . It’s the altered flavor of the soup that results from the chef’s culinary experiments. This variable depends on the changes made to the independent variable, hence the name!
Observing how the dependent variable reacts to changes helps scientists draw conclusions and make discoveries.
Control Variable Control variables are the unsung heroes of scientific research. They’re the constants, the elements that researchers keep the same to ensure the integrity of the experiment.
Imagine if our chef used a different type of broth each time he experimented with spices—the results would be all over the place! Control variables keep the experiment grounded and help researchers be confident in their findings.
Confounding Variables Imagine a hidden rock in a stream, changing the water’s flow in unexpected ways. Confounding variables are similar—they are external factors that can sneak into experiments and influence the outcome , adding twists to our scientific story.
These variables can blur the relationship between the independent and dependent variables, making the results of the study a bit puzzly. Detecting and controlling these hidden elements helps researchers ensure the accuracy of their findings and reach true conclusions.
There are of course other types of variables, and different ways to manipulate them called " schedules of reinforcement ," but we won't get into that too much here.
Manipulation When researchers manipulate the independent variable, they are orchestrating a symphony of cause and effect. They’re adjusting the strings, the brass, the percussion, observing how each change influences the melody—the dependent variable.
This manipulation is at the heart of experimental research. It allows scientists to explore relationships, unravel patterns, and unearth the secrets hidden within the fabric of our universe.
Observation With every tweak and adjustment made to the independent variable, researchers are like seasoned detectives, observing the dependent variable for changes, collecting clues, and piecing together the puzzle.
Observing the effects and changes that occur helps them deduce relationships, formulate theories, and expand our understanding of the world. Every observation is a step towards solving the mysteries of nature and human behavior.
Characteristics Identifying an independent variable in the vast landscape of research can seem daunting, but fear not! Independent variables have distinctive characteristics that make them stand out.
They’re the elements that are deliberately changed or controlled in an experiment to study their effects on the dependent variable. Recognizing these characteristics is like learning to spot footprints in the sand—it leads us to the heart of the discovery!
In Different Types of Research The world of research is diverse and varied, and the independent variable dons many guises! In the field of medicine, it might manifest as the dosage of a drug administered to patients.
In psychology, it could take the form of different learning methods applied to study memory retention. In each field, identifying the independent variable correctly is the golden key that unlocks the treasure trove of knowledge and insights.
As we forge ahead on our enlightening journey, equipped with a deeper understanding of independent variables and their roles, we’re ready to delve into the intricate theories and diverse examples that underscore their significance.
Now that we’re acquainted with the basic concepts and have the tools to identify independent variables, let’s dive into the fascinating ocean of theories and frameworks.
These theories are like ancient scrolls, providing guidelines and blueprints that help scientists use independent variables to uncover the secrets of the universe.
What is it and How Does it Work? The scientific method is like a super-helpful treasure map that scientists use to make discoveries. It has steps we follow: asking a question, researching, guessing what will happen (that's a hypothesis!), experimenting, checking the results, figuring out what they mean, and telling everyone about it.
Our hero, the independent variable, is the compass that helps this adventure go the right way!
How Independent Variables Lead the Way In the scientific method, the independent variable is like the captain of a ship, leading everyone through unknown waters.
Scientists change this variable to see what happens and to learn new things. It’s like having a compass that points us towards uncharted lands full of knowledge!
The Basics of Building Constructing an experiment is like building a castle, and the independent variable is the cornerstone. It’s carefully chosen and manipulated to see how it affects the dependent variable. Researchers also identify control and confounding variables, ensuring the castle stands strong, and the results are reliable.
Keeping Everything in Check In every experiment, maintaining control is key to finding the treasure. Scientists use control variables to keep the conditions consistent, ensuring that any changes observed are truly due to the independent variable. It’s like ensuring the castle’s foundation is solid, supporting the structure as it reaches for the sky.
Making Educated Guesses Before they start experimenting, scientists make educated guesses called hypotheses . It’s like predicting which X marks the spot of the treasure! It often includes the independent variable and the expected effect on the dependent variable, guiding researchers as they navigate through the experiment.
Independent Variables in the Spotlight When testing these guesses, the independent variable is the star of the show! Scientists change and watch this variable to see if their guesses were right. It helps them figure out new stuff and learn more about the world around us!
Figuring Out Relationships After the experimenting is done, it’s time for scientists to crack the code! They use statistics to understand how the independent and dependent variables are related and to uncover the hidden stories in the data.
Experimenters have to be careful about how they determine the validity of their findings, which is why they use statistics. Something called "experimenter bias" can get in the way of having true (valid) results, because it's basically when the experimenter influences the outcome based on what they believe to be true (or what they want to be true!).
How Important are the Discoveries? Through statistical analysis, scientists determine the significance of their findings. It’s like discovering if the treasure found is made of gold or just shiny rocks. The analysis helps researchers know if the independent variable truly had an effect, contributing to the rich tapestry of scientific knowledge.
As we uncover more about how theories and frameworks use independent variables, we start to see how awesome they are in helping us learn more about the world. But we’re not done yet!
Up next, we’ll look at tons of examples to see how independent variables work their magic in different areas.
Independent variables take on many forms, showcasing their versatility in a range of experiments and studies. Let’s uncover how they act as the protagonists in numerous investigations and learning quests!
1) plant growth.
Consider an experiment aiming to observe the effect of varying water amounts on plant height. In this scenario, the amount of water given to the plants is the independent variable!
Suppose we are curious about the time it takes for water to freeze at different temperatures. The temperature of the freezer becomes the independent variable as we adjust it to observe the results!
Have you ever observed how shadows change? In an experiment, adjusting the light angle to observe its effect on an object’s shadow makes the angle of light the independent variable!
In medical studies, determining how varying medicine dosages influence a patient’s recovery is essential. Here, the dosage of the medicine administered is the independent variable!
Researchers might examine the impact of different exercise forms on individuals’ health. The various exercise forms constitute the independent variable in this study!
Have you pondered how the sleep duration affects your well-being the following day? In such research, the hours of sleep serve as the independent variable!
Psychologists might investigate how diverse study methods influence test outcomes. Here, the different study methods adopted by students are the independent variable!
Have you experienced varied emotions with different music genres? The genre of music played becomes the independent variable when researching its influence on emotions!
Suppose researchers are exploring how room colors affect individuals’ emotions. In this case, the room colors act as the independent variable!
10) rainfall and plant life.
Environmental scientists may study the influence of varying rainfall levels on vegetation. In this instance, the amount of rainfall is the independent variable!
Examining how temperature variations affect animal behavior is fascinating. Here, the varying temperatures serve as the independent variable!
Investigating the effects of different pollution levels on air quality is crucial. In such studies, the pollution level is the independent variable!
Researchers might explore how varying internet speeds impact work productivity. In this exploration, the internet speed is the independent variable!
Examining how different devices affect user experience is interesting. Here, the type of device used is the independent variable!
Suppose a study aims to determine how different software versions influence system performance. The software version becomes the independent variable!
Educators might investigate the effect of varied teaching styles on student engagement. In such a study, the teaching style is the independent variable!
Researchers could explore how different class sizes influence students’ learning. Here, the class size is the independent variable!
Examining the relationship between the frequency of homework assignments and academic success is essential. The frequency of homework becomes the independent variable!
Astronomers might study how different telescopes affect celestial observation. In this scenario, the telescope type is the independent variable!
Investigating the influence of varying light pollution levels on star visibility is intriguing. Here, the level of light pollution is the independent variable!
Suppose a study explores how observation duration affects the detail captured in astronomical images. The duration of observation serves as the independent variable!
Sociologists may examine how the size of a community influences social interactions. In this research, the community size is the independent variable!
Investigating the effect of diverse cultural exposure on social tolerance is vital. Here, the level of cultural exposure is the independent variable!
Researchers could explore how different economic statuses impact educational achievements. In such studies, economic status is the independent variable!
Sports scientists might study how varying training intensities affect athletes’ performance. In this case, the training intensity is the independent variable!
Examining the relationship between different sports equipment and player safety is crucial. Here, the type of equipment used is the independent variable!
Suppose researchers are investigating how the size of a sports team influences game strategy. The team size becomes the independent variable!
Nutritionists may explore the impact of various diets on individuals’ health. In this exploration, the type of diet followed is the independent variable!
Investigating how different caloric intakes influence weight change is essential. In such a study, the caloric intake is the independent variable!
Researchers could examine how consuming a variety of foods affects nutrient absorption. Here, the variety of foods consumed is the independent variable!
Isn't it fantastic how independent variables play such an essential part in so many studies? But the excitement doesn't stop there!
Now, let’s explore how findings from these studies, led by independent variables, make a big splash in the real world and improve our daily lives!
31) treatment optimization.
By studying different medicine dosages and treatment methods as independent variables, doctors can figure out the best ways to help patients recover quicker and feel better. This leads to more effective medicines and treatment plans!
Researching the effects of sleep, exercise, and diet helps health experts give us advice on living healthier lives. By changing these independent variables, scientists uncover the secrets to feeling good and staying well!
33) speeding up the internet.
When scientists explore how different internet speeds affect our online activities, they’re able to develop technologies to make the internet faster and more reliable. This means smoother video calls and quicker downloads!
By examining how we interact with various devices and software, researchers can design technology that’s easier and more enjoyable to use. This leads to cooler gadgets and more user-friendly apps!
35) enhancing learning.
Investigating different teaching styles, class sizes, and study methods helps educators discover what makes learning fun and effective. This research shapes classrooms, teaching methods, and even homework!
By studying how students with diverse needs respond to different support strategies, educators can create personalized learning experiences. This means every student gets the help they need to succeed!
37) conserving nature.
Researching how rainfall, temperature, and pollution affect the environment helps scientists suggest ways to protect our planet. By studying these independent variables, we learn how to keep nature healthy and thriving!
Scientists studying the effects of pollution and human activities on climate change are leading the way in finding solutions. By exploring these independent variables, we can develop strategies to combat climate change and protect the Earth!
39) building stronger communities.
Sociologists studying community size, cultural exposure, and economic status help us understand what makes communities happy and united. This knowledge guides the development of policies and programs for stronger societies!
By exploring how exposure to diverse cultures affects social tolerance, researchers contribute to fostering more inclusive and harmonious societies. This helps build a world where everyone is respected and valued!
41) optimizing athlete training.
Sports scientists studying training intensity, equipment type, and team size help athletes reach their full potential. This research leads to better training programs, safer equipment, and more exciting games!
By investigating how different game strategies are influenced by various team compositions, researchers contribute to the evolution of sports. This means more thrilling competitions and matches for us to enjoy!
43) guiding healthy eating.
Nutritionists researching diet types, caloric intake, and food variety help us understand what foods are best for our bodies. This knowledge shapes dietary guidelines and helps us make tasty, yet nutritious, meal choices!
By studying the effects of different nutrients and diets, researchers educate us on maintaining a balanced diet. This fosters a greater awareness of nutritional well-being and encourages healthier eating habits!
As we journey through these real-world applications, we witness the incredible impact of studies featuring independent variables. The exploration doesn’t end here, though!
Let’s continue our adventure and see how we can identify independent variables in our own observations and inquiries! Keep your curiosity alive, and let’s delve deeper into the exciting realm of independent variables!
So, we’ve seen how independent variables star in many studies, but how about spotting them in our everyday life?
Recognizing independent variables can be like a treasure hunt – you never know where you might find one! Let’s uncover some tips and tricks to identify these hidden gems in various situations.
One of the best ways to spot an independent variable is by asking questions! If you’re curious about something, ask yourself, “What am I changing or manipulating in this situation?” The thing you’re changing is likely the independent variable!
For example, if you’re wondering whether the amount of sunlight affects how quickly your laundry dries, the sunlight amount is your independent variable!
Keep your eyes peeled and observe the world around you! By watching how changes in one thing (like the amount of rain) affect something else (like the height of grass), you can identify the independent variable.
In this case, the amount of rain is the independent variable because it’s what’s changing!
Get hands-on and conduct your own experiments! By changing one thing and observing the results, you’re identifying the independent variable.
If you’re growing plants and decide to water each one differently to see the effects, the amount of water is your independent variable!
In everyday scenarios, independent variables are all around!
When you adjust the temperature of your oven to bake cookies, the oven temperature is the independent variable.
Or if you’re deciding how much time to spend studying for a test, the study time is your independent variable!
Keep being curious and asking “What if?” questions! By exploring different possibilities and wondering how changing one thing could affect another, you’re on your way to identifying independent variables.
If you’re curious about how the color of a room affects your mood, the room color is the independent variable!
Don’t forget about the treasure trove of past studies and experiments! By reviewing what scientists and researchers have done before, you can learn how they identified independent variables in their work.
This can give you ideas and help you recognize independent variables in your own explorations!
Ready for some practice? Let’s put on our thinking caps and try to identify the independent variables in a few scenarios.
Remember, the independent variable is what’s being changed or manipulated to observe the effect on something else! (You can see the answers below)
You’re cooking pasta for dinner and want to find out how the cooking time affects its texture. What is the independent variable?
You decide to try different exercise routines each week to see which one makes you feel the most energetic. What is the independent variable?
You’re growing tomatoes in your garden and decide to use different types of fertilizer to see which one helps them grow the best. What is the independent variable?
You’re preparing for an important test and try studying in different environments (quiet room, coffee shop, library) to see where you concentrate best. What is the independent variable?
You’re curious to see how the number of hours you sleep each night affects your mood the next day. What is the independent variable?
By practicing identifying independent variables in different scenarios, you’re becoming a true independent variable detective. Keep practicing, stay curious, and you’ll soon be spotting independent variables everywhere you go.
Independent Variable: The cooking time is the independent variable. You are changing the cooking time to observe its effect on the texture of the pasta.
Independent Variable: The type of exercise routine is the independent variable. You are trying out different exercise routines each week to see which one makes you feel the most energetic.
Independent Variable: The type of fertilizer is the independent variable. You are using different types of fertilizer to observe their effects on the growth of the tomatoes.
Independent Variable: The study environment is the independent variable. You are studying in different environments to see where you concentrate best.
Independent Variable: The number of hours you sleep is the independent variable. You are changing your sleep duration to see how it affects your mood the next day.
Whew, what a journey we’ve had exploring the world of independent variables! From understanding their definition and role to diving into a myriad of examples and real-world impacts, we’ve uncovered the treasures hidden in the realm of independent variables.
The beauty of independent variables lies in their ability to unlock new knowledge and insights, guiding us to discoveries that improve our lives and the world around us.
By identifying and studying these variables, we embark on exciting learning adventures, solving mysteries and answering questions about the universe we live in.
Remember, the joy of discovery doesn’t end here. The world is brimming with questions waiting to be answered and mysteries waiting to be solved.
Keep your curiosity alive, continue exploring, and who knows what incredible discoveries lie ahead.
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Published on 4 May 2022 by Pritha Bhandari . Revised on 17 October 2022.
In research, variables are any characteristics that can take on different values, such as height, age, temperature, or test scores.
Researchers often manipulate or measure independent and dependent variables in studies to test cause-and-effect relationships.
Your independent variable is the temperature of the room. You vary the room temperature by making it cooler for half the participants, and warmer for the other half.
What is an independent variable, types of independent variables, what is a dependent variable, identifying independent vs dependent variables, independent and dependent variables in research, visualising independent and dependent variables, frequently asked questions about independent and dependent variables.
An independent variable is the variable you manipulate or vary in an experimental study to explore its effects. It’s called ‘independent’ because it’s not influenced by any other variables in the study.
Independent variables are also called:
These terms are especially used in statistics , where you estimate the extent to which an independent variable change can explain or predict changes in the dependent variable.
There are two main types of independent variables.
In experiments, you manipulate independent variables directly to see how they affect your dependent variable. The independent variable is usually applied at different levels to see how the outcomes differ.
You can apply just two levels in order to find out if an independent variable has an effect at all.
You can also apply multiple levels to find out how the independent variable affects the dependent variable.
You have three independent variable levels, and each group gets a different level of treatment.
You randomly assign your patients to one of the three groups:
A true experiment requires you to randomly assign different levels of an independent variable to your participants.
Random assignment helps you control participant characteristics, so that they don’t affect your experimental results. This helps you to have confidence that your dependent variable results come solely from the independent variable manipulation.
Subject variables are characteristics that vary across participants, and they can’t be manipulated by researchers. For example, gender identity, ethnicity, race, income, and education are all important subject variables that social researchers treat as independent variables.
It’s not possible to randomly assign these to participants, since these are characteristics of already existing groups. Instead, you can create a research design where you compare the outcomes of groups of participants with characteristics. This is a quasi-experimental design because there’s no random assignment.
Your independent variable is a subject variable, namely the gender identity of the participants. You have three groups: men, women, and other.
Your dependent variable is the brain activity response to hearing infant cries. You record brain activity with fMRI scans when participants hear infant cries without their awareness.
A dependent variable is the variable that changes as a result of the independent variable manipulation. It’s the outcome you’re interested in measuring, and it ‘depends’ on your independent variable.
In statistics , dependent variables are also called:
The dependent variable is what you record after you’ve manipulated the independent variable. You use this measurement data to check whether and to what extent your independent variable influences the dependent variable by conducting statistical analyses.
Based on your findings, you can estimate the degree to which your independent variable variation drives changes in your dependent variable. You can also predict how much your dependent variable will change as a result of variation in the independent variable.
Distinguishing between independent and dependent variables can be tricky when designing a complex study or reading an academic paper.
A dependent variable from one study can be the independent variable in another study, so it’s important to pay attention to research design.
Here are some tips for identifying each variable type.
Use this list of questions to check whether you’re dealing with an independent variable:
Check whether you’re dealing with a dependent variable:
Independent and dependent variables are generally used in experimental and quasi-experimental research.
Here are some examples of research questions and corresponding independent and dependent variables.
Research question | Independent variable | Dependent variable(s) |
---|---|---|
Do tomatoes grow fastest under fluorescent, incandescent, or natural light? | ||
What is the effect of intermittent fasting on blood sugar levels? | ||
Is medical marijuana effective for pain reduction in people with chronic pain? | ||
To what extent does remote working increase job satisfaction? |
For experimental data, you analyse your results by generating descriptive statistics and visualising your findings. Then, you select an appropriate statistical test to test your hypothesis .
The type of test is determined by:
You’ll often use t tests or ANOVAs to analyse your data and answer your research questions.
In quantitative research , it’s good practice to use charts or graphs to visualise the results of studies. Generally, the independent variable goes on the x -axis (horizontal) and the dependent variable on the y -axis (vertical).
The type of visualisation you use depends on the variable types in your research questions:
To inspect your data, you place your independent variable of treatment level on the x -axis and the dependent variable of blood pressure on the y -axis.
You plot bars for each treatment group before and after the treatment to show the difference in blood pressure.
An independent variable is the variable you manipulate, control, or vary in an experimental study to explore its effects. It’s called ‘independent’ because it’s not influenced by any other variables in the study.
A dependent variable is what changes as a result of the independent variable manipulation in experiments . It’s what you’re interested in measuring, and it ‘depends’ on your independent variable.
In statistics, dependent variables are also called:
Determining cause and effect is one of the most important parts of scientific research. It’s essential to know which is the cause – the independent variable – and which is the effect – the dependent variable.
You want to find out how blood sugar levels are affected by drinking diet cola and regular cola, so you conduct an experiment .
Yes, but including more than one of either type requires multiple research questions .
For example, if you are interested in the effect of a diet on health, you can use multiple measures of health: blood sugar, blood pressure, weight, pulse, and many more. Each of these is its own dependent variable with its own research question.
You could also choose to look at the effect of exercise levels as well as diet, or even the additional effect of the two combined. Each of these is a separate independent variable .
To ensure the internal validity of an experiment , you should only change one independent variable at a time.
No. The value of a dependent variable depends on an independent variable, so a variable cannot be both independent and dependent at the same time. It must be either the cause or the effect, not both.
If you want to cite this source, you can copy and paste the citation or click the ‘Cite this Scribbr article’ button to automatically add the citation to our free Reference Generator.
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Statistics By Jim
Making statistics intuitive
By Jim Frost 14 Comments
When comparing groups in your data, you can have either independent or dependent samples. The type of samples in your experimental design impacts sample size requirements, statistical power, the proper analysis, and even your study’s costs. Understanding the implications of each type of sample can help you design a better experiment.
In this post, I’ll define independent and dependent samples, explain their pros and cons, highlight the appropriate analyses for each type, and illustrate how dependent groups can increase your statistical power.
A quick note about terminology. In experiments, you measure an outcome variable for people or objects. I’ll refer to subjects throughout this post to refer to both cases. Additionally, I also use samples and groups synonymously. For example, the term “dependent samples” means the same thing as dependent groups.
Hypothesis tests and statistical modeling that compare groups have assumptions about the nature of those groups. Choosing the correct test or model depends on knowing which type of groups your experiment has. Additionally, when designing your study, selecting the best type can help you tailor the design to meet your needs.
In independent samples, subjects in one group do not provide information about subjects in other groups. Each group contains different subjects and there is no meaningful way to pair them. Independent groups are more common in hypothesis testing.
For example, the following experiments use independent samples:
Studies that use independent samples estimate between-subject effects. These effects are the differences between groups, such as the mean difference. For example, in the medication study, the effect is the mean difference between the treatment and control groups. The focus is on comparing group properties rather than individuals. The sample size for this type of study is the total number of subjects in all groups.
Related post : Independent Samples T Test
Groups are frequently dependent because they contain the same subjects—that’s the most common example. However, that’s not always the case. Groups with different subjects can be dependent samples if the subjects in one group provide information about the subjects in the other group. For example, statisticians often consider different samples that include pairs of siblings to be dependent because one sibling can provide information about another sibling for some measurements. Other studies use matched pairs. In these studies, the researchers deliberately pair subjects with very similar characteristics. While matched pairs are different people, the statistical analysis treats them as the same person because they are intentionally very similar.
For example, the following experiments use dependent samples:
Studies that use dependent samples estimate within-subject effects. These effects are the differences between paired subjects, such as the subjects’ mean change. For example, the training program assessment estimates the mean change for subjects from the pretest to the posttest. The emphasis is on the differences between paired subjects. The sample size for this type of study is the number of pairs.
Terms such as paired, repeated measurements, within-subject effects, matched pairs, and pretest/posttest indicate that the groups are dependent.
Related post : Paired T Test
Understanding how researchers record the data can also provide hints about the types of groups. For example, the data look similar in the two worksheets below.
For dependent groups, the focus is on the differences between measurements for each subject. Consequently, if you can meaningfully subtract values in a row, that’s a sure sign of dependency. For example, each row represents one individual in the paired dataset, so assessing the difference between values makes sense.
Conversely, for the independent samples dataset, each group contains a different set of individuals that the researchers chose randomly. Each row in this dataset does not pertain to a single subject. Consequently, it does not make sense to subtract the values between pairs of random people.
When thinking about comparing groups, you frequently picture independent groups. For instance, when you imagine comparing a treatment group to a control group, you’re probably assuming these groups contain different subjects. However, by understanding the pros and cons of independent and dependent samples, you can design a study to meet your needs more effectively. The best choice depends on the subject matter and requirements of your experiment. Consider the following while deciding your approach.
When your study uses independent samples, you test each subject once. When you’re working with human subjects, a single test can be advantageous for several reasons. With a single assessment per person, you don’t need to worry about subjects learning how to perform better, getting bored with multiple tests, and how the passage of time affects each person. By testing subjects once, you can rule out various time and order effects that can influence how scores change.
When you are testing physical items, you only need to test each item once. If the testing damages or alters the items, it’s not possible to test them multiple times.
Because each group contains different subjects, there can be a wide variety of subject specific factors that influence how they respond to the test. While random assignment to groups can reduce systematic differences between groups, these subject specific factors are not controlled.
Differences between participants in the groups can affect the results. Statisticians refer to these differences as participant variables and they include age, gender, and social background, among many other possibilities.
The additional variability that participant variables create reduces statistical power. You generally need larger sample sizes with independent samples.
The primary advantage of dependent samples is that you measure the same subjects across different conditions, which allows them to be their own controls. They have the same unique mix of participant variables during all measurements, removing them as sources of variation. Keep this lower variability in mind during my practical demonstration later in this post!
For example, in a pretest/posttest analysis, you will see how each subject reacts to both tests. This method allows the study to focus on the changes within individuals rather than differences between groups of different people.
The net effect is a gain in statistical power. You generally need smaller sample sizes with dependent groups. Additionally, reducing the sample size can decrease a study’s costs, which is particularly helpful when it is difficult or expensive to obtain subjects.
When working with human subjects, you will need to test them multiple times with dependent samples. During repeated testing, subjects can learn more about the tests and figure out how to improve their scores; they might get bored with being tested multiple times; or their test scores might change as a natural result of time passing. In other words, the multiple testing and the passage of time become factors than can influence the measurement, potentially making it challenging to isolate the treatment’s effect.
For example, if the test scores for the training program increase from the pretest to the posttest, the training program might not cause the change. Instead, participants might be learning how to take the test better!
Researchers can mitigate some of these problems. For example, they can include control groups for comparison and change the order of tests for subsets of subjects. However, in general, designs that use dependent groups make it easier for alternatives to explain the changes.
In some cases, using dependent samples is not possible. For example, with destructive testing of material objects, you can only test them once!
As a researcher, weigh the benefits and drawbacks of both types of samples. Some types of research will lend themselves to one approach or the other.
After choosing the type of samples and conducting the experiment, you need to use the correct statistical analysis. The table displays pairs of related analyses for independent and dependent samples.
While analyses for dependent groups typically focus on individual changes, McNemar’s test is an exception. That test compares the overall proportions of two dependent groups.
Regression and ANOVA can model both independent and dependent samples. It’s just a matter of specifying the correct model.
Related posts : Repeated Measures ANOVA and How to do t-tests
I’m closing with an example that illustrates the extra statistical power that dependent samples can provide. Imagine two studies that, by an amazing coincidence, obtain the same measurements exactly. The only difference is that one has independent groups, while the other has dependent groups.
It should go without saying, but I’ll say it anyway—you will never run a 2-sample t-test and a paired t-test on the same dataset in practice. The two designs are entirely incompatible. However, I’m going to do just that to illustrate the difference in power.
For this experiment, we’re assessing a fictional drug that supposedly increases IQ scores. One experiment uses a control group and a treatment group that have different subjects. The other uses the same set of subjects for a pretest and a posttest. You can download the CSV dataset to try it yourself: IndDepSamples .
First, let’s analyze the dataset as a 2-sample t-test.
Ok, now let’s use the paired t-test.
The data are the same for both analyses and the differences between samples are the same (-11.62). The 2-sample t-test uses a sample size of 30 (two groups with 15 per group), while the paired t-test has only 15 subjects, but the researchers test them twice. Why is the paired t-test with the dependent samples statistically significant while the 2-sample t-test with independent samples is not significant?
The analyses make different assumptions about the nature of the samples. For the 2-sample t-test, the two groups contain entirely different individuals. While the treatment group has a higher mean IQ score than the control group, we don’t know each subject’s starting score because there was no pretest. Perhaps the treatment group started with higher scores by chance? We don’t know for sure if anyone’s scores increased after taking the drug. This uncertainty reduces the test’s power.
On the other hand, the paired t-test assumes that the pretest and posttest scores are from the same people. From the data, we know all 15 participants saw their scores increase from the pretest to the posttest by an average of 11.63 points. That’s a pretty powerful contrast to the independent samples where we don’t know if any IQ scores increased during the study. While we can be reasonably confident that their scores increased, we’re not sure why. It’s possible that their experience taking the pretest helped them do better on the posttest. Tradeoffs! Maybe next time we’ll include a control group and perform repeated measures ANOVA.
For a more statistical explanation, think back to what I said about dependent samples eliminating participant variables as a source of variability. You can see the reduced variability in the statistical output. The 2-sample t-test uses the pooled standard deviation for both groups, which the output indicates is about 19. However, the paired t-test uses the standard deviation of the differences , and that is much lower at only 6.81. In t-tests, variability is noise that can obscure the signal. Consequently, higher variability reduces statistical power. For more information on this aspect, read my post about how t-tests work .
If you’re planning your next study, consider whether you should use independent or dependent samples. Throughout this post, you learned that each approach has its own benefits and drawbacks. Determine which one works best for your study.
Read more about the related topic of independent and identically distributed (IID) data .
May 26, 2021 at 5:07 am
Hello,Jim, thank you for posting this article. After reading this article, I am thinking that maybe you can help to answer my question. My question is how to determine the correct sample size for dependent sampling. looking forward to your reply. Thanks again~
May 26, 2021 at 3:52 pm
You need to perform a power and sample size analysis . Click the link to learn more. This process helps you determine the correct sample size. In your statistical software, you’ll need to specify an analysis appropriate for dependent samples, such as a paired t-test.
I hope that helps!
February 20, 2021 at 8:16 am
Thanks for this post. I am trying to figure out what my sample numbers should be. I am testing team input/process variables to see which correlate with team cohesion. My questionnaire is to multicultural teams. I originally had in mind 30 like that of good grounded theory. My goal is to see what variables cohesive multicultural teams have. I want to make a meaningful contribution with the research. I am surveying missionary teams as they are commonly multicultural and I have access. I am using these teams from multiple organizations in order to get a good representation in my sample. Please give me a little guidance, which I think will help others who read this. I’ve had several others who are looking to do this type of stats/research, and we are all too statistically novice to know what to do. Thanks, in advance.
October 13, 2020 at 11:52 pm
Here’s some questions;
Statistics are used especially in psychology, sociology and economics. Why? Consider, in psychology, why it is paired with experimental method
October 14, 2020 at 8:57 pm
For your question, I’m going to assume you’re referring to inferential statistics because those methods really extend the usefulness of experiments.
Inferential statistics are a set of analyses that allow you to use sample data to draw conclusions about an entire population. These procedures are very important for scientific studies, including in the areas you mention. Imagine a psychology study that is looking at a treatment for a psychological condition. For this study, the scientists will gather a sample of study participants. The scientists don’t want to know whether the treatment works for only this relatively small group of people in the study. That’s not very helpful for everyone else! Instead, these scientists want to understand how the treatment will work in a larger population. They can use inferential statistics to take the results from their sample and generalize them to an entire population. That makes their study much more useful!
By pairing these statistical procedures with experiments, it allows the researchers to draw conclusions about how effective the treatment is for an entire population, not just the small group of subjects in the experiment.
Thanks for writing with the great question!
October 13, 2020 at 11:51 pm
Statistics are often seen as untrustworthy, and used to prove whatever a person may want to prove. What are some of the common suspicions about the use of statistics?
October 14, 2020 at 8:27 pm
Whenever I read about someone’s statistical analysis, my first concern is about how they collected their data. Did they collect their data from a group of friends who already share their opinion? Or, did they randomly sample people? Data collecting can completely change the results of the analysis.
After data collection, I’d want to understand the specifics about how they analyzed their data. Many analyses can be twisted or misused to give whatever answer the person wants. However, if you collect data properly and use analyses properly, statistics tend to give the correct answers. But, it’s important to understand all the details about how they arrive at their conclusions.
Finally, the best way to protect yourself from someone else misusing statistics is to become knowledgeable in statistics yourself. By understanding statistics, you’ll be able to know what to look for to see through someone else’s statistical trickery!
Thanks for writing and best wishes!
September 7, 2020 at 11:42 am
Thank you Jim. I have benefited from your article
September 2, 2020 at 8:06 pm
Thanks for that Jim. I will give the article a read with that in mind. Tony
September 2, 2020 at 11:49 am
Thank you, Jim. This is an excellent refresher on t-tests and introduces new terminology, i.e., dependent and independent samples. It’s always good to look “under the hood” and see how things work.
September 2, 2020 at 6:12 am
Hi Jim, before I try to wade through your lengthy article, right away I feel this is not the way my study of statistics is going.
My direction, post college statistics, is now data science, machine learning supervised learning alogorithms, as with logistic regression. What little I’ve learned about unsupervised learning is a single project using k-means clustering.
What I know about data sets are pre-processing, data wrangling, modeling and cross validation.
Part of the data wrangling process includes choosing among variables a dependent target variable.
When I saw your title Dependent Samples, I need to understand what is the value here for me.
Let me guess you write for purposes of medical research, not analytics in business.
Thank you for your time.
September 2, 2020 at 2:51 pm
Thanks for your thoughts. Every time I write a blog post, I have no doubt that it will be more helpful for some and less helpful for others. Everyone has their own unique needs. Such is life. I’m sorry this post didn’t help you specifically, but I’m sure it helped others.
I write about all sorts of topics that will be helpful for people learning statistics in a broad range of contexts, including business and machine learning. In this post, some of the content focuses on issues for designing experiments. While that seems to be relevant to scientific fields, many businesses also design experiments. Additionally, I’m sure businesses collect multiple measurements on subjects. This post addresses how that affects the analyses you must use, how to interpret the results, as well as the benefits and risks in terms of explaining the results. This information should be useful in many contexts, such as businesses.
At some point, I plan to write about analyses such as k-means clustering.
Finally, every time I write a post, I include what it’ll be about and the benefits of reading the post right at the beginning. This information allows everyone to decide for themselves if they should read it.
September 2, 2020 at 4:22 am
Thank You for such a great effort in the area of statistics. Well done! Keep it up! Best Wishes!
September 2, 2020 at 1:58 am
Thank you very much Mr. Jim for your effort of sharing your knowledge about the concepts of statistics area.
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Published on March 20, 2020 by Rebecca Bevans . Revised on June 22, 2023.
ANOVA (Analysis of Variance) is a statistical test used to analyze the difference between the means of more than two groups.
A two-way ANOVA is used to estimate how the mean of a quantitative variable changes according to the levels of two categorical variables. Use a two-way ANOVA when you want to know how two independent variables, in combination, affect a dependent variable.
When to use a two-way anova, how does the anova test work, assumptions of the two-way anova, how to perform a two-way anova, interpreting the results of a two-way anova, how to present the results of a a two-way anova, other interesting articles, frequently asked questions about two-way anova.
You can use a two-way ANOVA when you have collected data on a quantitative dependent variable at multiple levels of two categorical independent variables.
A quantitative variable represents amounts or counts of things. It can be divided to find a group mean.
A categorical variable represents types or categories of things. A level is an individual category within the categorical variable.
You should have enough observations in your data set to be able to find the mean of the quantitative dependent variable at each combination of levels of the independent variables.
Both of your independent variables should be categorical. If one of your independent variables is categorical and one is quantitative, use an ANCOVA instead.
ANOVA tests for significance using the F test for statistical significance . The F test is a groupwise comparison test, which means it compares the variance in each group mean to the overall variance in the dependent variable.
If the variance within groups is smaller than the variance between groups, the F test will find a higher F value, and therefore a higher likelihood that the difference observed is real and not due to chance.
A two-way ANOVA with interaction tests three null hypotheses at the same time:
A two-way ANOVA without interaction (a.k.a. an additive two-way ANOVA) only tests the first two of these hypotheses.
Null hypothesis (H ) | Alternate hypothesis (H ) |
---|---|
There is no difference in average yield for any fertilizer type. | There is a difference in average yield by fertilizer type. |
There is no difference in average yield at either planting density. | There is a difference in average yield by planting density. |
The effect of one independent variable on average yield does not depend on the effect of the other independent variable (a.k.a. no interaction effect). | There is an interaction effect between planting density and fertilizer type on average yield. |
To use a two-way ANOVA your data should meet certain assumptions.Two-way ANOVA makes all of the normal assumptions of a parametric test of difference:
The variation around the mean for each group being compared should be similar among all groups. If your data don’t meet this assumption, you may be able to use a non-parametric alternative , like the Kruskal-Wallis test.
Your independent variables should not be dependent on one another (i.e. one should not cause the other). This is impossible to test with categorical variables – it can only be ensured by good experimental design .
In addition, your dependent variable should represent unique observations – that is, your observations should not be grouped within locations or individuals.
If your data don’t meet this assumption (i.e. if you set up experimental treatments within blocks), you can include a blocking variable and/or use a repeated-measures ANOVA.
The values of the dependent variable should follow a bell curve (they should be normally distributed ). If your data don’t meet this assumption, you can try a data transformation.
The dataset from our imaginary crop yield experiment includes observations of:
The two-way ANOVA will test whether the independent variables (fertilizer type and planting density) have an effect on the dependent variable (average crop yield). But there are some other possible sources of variation in the data that we want to take into account.
We applied our experimental treatment in blocks, so we want to know if planting block makes a difference to average crop yield. We also want to check if there is an interaction effect between two independent variables – for example, it’s possible that planting density affects the plants’ ability to take up fertilizer.
Because we have a few different possible relationships between our variables, we will compare three models:
Model 1 assumes there is no interaction between the two independent variables. Model 2 assumes that there is an interaction between the two independent variables. Model 3 assumes there is an interaction between the variables, and that the blocking variable is an important source of variation in the data.
By running all three versions of the two-way ANOVA with our data and then comparing the models, we can efficiently test which variables, and in which combinations, are important for describing the data, and see whether the planting block matters for average crop yield.
This is not the only way to do your analysis, but it is a good method for efficiently comparing models based on what you think are reasonable combinations of variables.
We will run our analysis in R. To try it yourself, download the sample dataset.
Sample dataset for a two-way ANOVA
After loading the data into the R environment, we will create each of the three models using the aov() command, and then compare them using the aictab() command. For a full walkthrough, see our guide to ANOVA in R .
This first model does not predict any interaction between the independent variables, so we put them together with a ‘+’.
In the second model, to test whether the interaction of fertilizer type and planting density influences the final yield, use a ‘ * ‘ to specify that you also want to know the interaction effect.
Because our crop treatments were randomized within blocks, we add this variable as a blocking factor in the third model. We can then compare our two-way ANOVAs with and without the blocking variable to see whether the planting location matters.
Now we can find out which model is the best fit for our data using AIC ( Akaike information criterion ) model selection.
AIC calculates the best-fit model by finding the model that explains the largest amount of variation in the response variable while using the fewest parameters. We can perform a model comparison in R using the aictab() function.
The output looks like this:
The AIC model with the best fit will be listed first, with the second-best listed next, and so on. This comparison reveals that the two-way ANOVA without any interaction or blocking effects is the best fit for the data.
You can view the summary of the two-way model in R using the summary() command. We will take a look at the results of the first model, which we found was the best fit for our data.
The model summary first lists the independent variables being tested (‘fertilizer’ and ‘density’). Next is the residual variance (‘Residuals’), which is the variation in the dependent variable that isn’t explained by the independent variables.
The following columns provide all of the information needed to interpret the model:
From this output we can see that both fertilizer type and planting density explain a significant amount of variation in average crop yield ( p values < 0.001).
ANOVA will tell you which parameters are significant, but not which levels are actually different from one another. To test this we can use a post-hoc test. The Tukey’s Honestly-Significant-Difference (TukeyHSD) test lets us see which groups are different from one another.
This output shows the pairwise differences between the three types of fertilizer ($fertilizer) and between the two levels of planting density ($density), with the average difference (‘diff’), the lower and upper bounds of the 95% confidence interval (‘lwr’ and ‘upr’) and the p value of the difference (‘p-adj’).
From the post-hoc test results, we see that there are significant differences ( p < 0.05) between:
but no difference between fertilizer groups 2 and 1.
Once you have your model output, you can report the results in the results section of your thesis , dissertation or research paper .
When reporting the results you should include the F statistic, degrees of freedom, and p value from your model output.
You can discuss what these findings mean in the discussion section of your paper.
You may also want to make a graph of your results to illustrate your findings.
Your graph should include the groupwise comparisons tested in the ANOVA, with the raw data points, summary statistics (represented here as means and standard error bars), and letters or significance values above the groups to show which groups are significantly different from the others.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
The only difference between one-way and two-way ANOVA is the number of independent variables . A one-way ANOVA has one independent variable, while a two-way ANOVA has two.
All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a t-test instead.
In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.
Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).
If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.
A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A two-way ANOVA is a type of factorial ANOVA.
Some examples of factorial ANOVAs include:
Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).
Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).
You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .
If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.
Bevans, R. (2023, June 22). Two-Way ANOVA | Examples & When To Use It. Scribbr. Retrieved August 21, 2024, from https://www.scribbr.com/statistics/two-way-anova/
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For example, adding a fourth independent variable with three levels (e.g., therapist experience: low vs. medium vs. high) to the current example would make it a 2 × 2 × 2 × 3 factorial design with 24 distinct conditions. In the rest of this section, we will focus on designs with two independent variables.
The study by Schnall and colleagues is a good example. One independent variable was disgust, which the researchers manipulated by testing participants in a clean room or a messy room. ... The results of factorial experiments with two independent variables can be graphed by representing one ... This led to the hypothesis that people high in ...
In this example, the independent variable is exposure to the sun - the assumed cause. ... If you are comparing two groups, the hypothesis can state what difference you expect to find between them. First-year students who attended most lectures will have better exam scores than those who attended few lectures. 6. Write a null hypothesis
The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable. It states results are due to chance and are not significant in supporting the idea being investigated.
A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process. Consider a study designed to examine the relationship between sleep deprivation and test ...
A simple hypothesis only predicts the relationship between one independent and another independent variable. Example: " Applying sunscreen every day slows skin aging." 6. Complex hypothesis: A complex hypothesis states the relationship or difference between two or more independent and dependent variables.
A simple hypothesis suggests only the relationship between two variables: one independent and one dependent. Examples: If you stay up late, then you feel tired the next day. Turning off your phone makes it charge faster. 2 Complex hypothesis. A complex hypothesis suggests the relationship between more than two variables, for example, two ...
While the independent variable is the " cause ", the dependent variable is the " effect " - or rather, the affected variable. In other words, the dependent variable is the variable that is assumed to change as a result of a change in the independent variable. Keeping with the previous example, let's look at some dependent variables ...
Step 5: Phrase your hypothesis in three ways. To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable. If a first-year student starts attending more lectures, then their exam scores will improve.
Independent Samples T Tests Hypotheses. Independent samples t tests have the following hypotheses: Null hypothesis: The means for the two populations are equal. Alternative hypothesis: The means for the two populations are not equal.; If the p-value is less than your significance level (e.g., 0.05), you can reject the null hypothesis. The difference between the two means is statistically ...
The expected frequencies are computed assuming that the null hypothesis is true. The null hypothesis states that the two variables (the grouping variable and the outcome) are independent. The definition of independence is as follows: Two events, A and B, are independent if P(A|B) = P(A), or equivalently, if P(A and B) = P(A) P(B).
The independent variable is the catalyst, the initial spark that sets the wheels of research in motion. Dependent Variable. The dependent variable is the outcome we observe and measure. It's the altered flavor of the soup that results from the chef's culinary experiments.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (HA): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...
To form a solid theory, the vital first step is creating a hypothesis. See the various types of hypotheses and how they can lead you on the path to discovery.
The independent variable is the cause. Its value is independent of other variables in your study. The dependent variable is the effect. Its value depends on changes in the independent variable. Example: Independent and dependent variables. You design a study to test whether changes in room temperature have an effect on math test scores.
Independent and Dependent Variables: Differences & Examples. By Jim Frost 15 Comments. Independent variables and dependent variables are the two fundamental types of variables in statistical modeling and experimental designs. Analysts use these methods to understand the relationships between the variables and estimate effect sizes.
Example: If you change the color of light (independent variable), then it affects plant growth (dependent variable). Example: If plant growth rate changes, then it affects the color of light. Sometimes you don't control either variable, like when you gather data to see if there is a relationship between two factors.
Calculations for two samples of data (both dependent or both independent) necessary to reject or accept the null hypothesis. Details.
A two sample t-test is used to determine whether or not two ... and 0.01) then you can reject the null hypothesis. Two Sample t-test: Assumptions. For the results of a two sample t-test to be valid, the following assumptions should be met: The observations in one sample should be independent of the observations in the other sample. The data ...
The formula for a multiple linear regression is: = the predicted value of the dependent variable. = the y-intercept (value of y when all other parameters are set to 0) = the regression coefficient () of the first independent variable () (a.k.a. the effect that increasing the value of the independent variable has on the predicted y value ...
The independent variable is the cause. Its value is independent of other variables in your study. The dependent variable is the effect. Its value depends on changes in the independent variable. Example: Independent and dependent variables. You design a study to test whether changes in room temperature have an effect on maths test scores.
The 2-sample t-test uses a sample size of 30 (two groups with 15 per group), while the paired t-test has only 15 subjects, but the researchers test them twice. Why is the paired t-test with the dependent samples statistically significant while the 2-sample t-test with independent samples is not significant? Understanding the Different Results
ANOVA (Analysis of Variance) is a statistical test used to analyze the difference between the means of more than two groups. A two-way ANOVA is used to estimate how the mean of a quantitative variable changes according to the levels of two categorical variables. Use a two-way ANOVA when you want to know how two independent variables, in ...