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AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 6.

• Statistical significance of experiment

Random sampling vs. random assignment (scope of inference)

• Conclusions in observational studies versus experiments
• Finding errors in study conclusions
• (Choice A)   Just the residents involved in Hilary's study. A Just the residents involved in Hilary's study.
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• (Choice A)   Yes A Yes
• (Choice B)   No B No
• (Choice A)   Just the residents in Hilary's study. A Just the residents in Hilary's study.

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34.4 - creating random assignments.

We now turn our focus from randomly sampling a subset of observations from a data set to that of generating a random assignment of treatments to experimental units in a randomized, controlled experiment. The good news is that the techniques used to sample without replacement can easily be extended to generate such random assignment plans.

It's probably a good time to remind you of the existence of the PLAN procedure. As I mentioned earlier, due to time constraints of the course and the complexity of the PLAN procedure, we will not use it to accomplish any of our random assignments. You should be aware, however, of its existence should you want to explore it on your own in the future.

Example 34.15 Section  

Suppose we are interested in conducting an experiment so that we can compare the effects of two drugs (A and B) and one placebo on headache pain. We have 30 subjects enrolled in our study but need to determine a plan for randomly assigning 10 of the subjects to treatment A, 10 of the subjects to treatment B, and 10 of the subjects to the placebo. The following program does just that for us. That is, it creates a random assignment for 30 subjects in a completely randomized design with one factor having 3 levels:

Okay, let's first launch and run    the SAS program, so you can review the resulting output to convince yourself that the code did indeed generate the desired treatment plan. You should see that 10 of the subjects were randomly assigned to treatment A, 10 to treatment B, and 10 to the placebo.

Now, let's walk ourselves through the program to make sure we understand how it works. The first DATA step merely uses a simple DO loop to create a temporary data set called exper1 that contains one observation for each of the experimental units (in our case, the experimental units are subjects). The only variable in the data set, unit , contains an arbitrary label 1, 2, ..., 30 assigned to each of the experimental units.

The remainder of the code generates a random assignment. To do so, the code from Example 34.5 is simply extended. That is:

• The second DATA step uses the ranuni function to generate a uniform random number between 0 and 1 for each observation in the exper1 data set. The result is stored in a temporary data set called random1 .
• The random1 data set is sorted in order of the random number.
• The third DATA step uses an IF-THEN-ELSE construct to assign the first ten units in sorted order to Group 1, the second ten to Group 2, and the last ten to Group 3.
• A FORMAT is defined to label the groups meaningfully.
• The final randomization list is printed.

Example 34.16 Section  

To create a random assignment for a completely randomized design with two factors , you can just modify the IF statement in the previous example. The following program generates a random assignment of treatments to 30 subjects, in which Factor A has 2 levels and Factor B has 3 levels (and hence 6 treatments). The code is similar to the code from the previous example except the IF statement now divides the 30 subjects into 6 treatment groups and (arbitrarily) assigns the levels of factors A and B to the groups:

First, my apologies for the formatting that makes the IF-THEN-ELSE statement a little difficult to read. I needed to format it as such so that I could easily capture the image of the program for you.

Again, it's probably best if you first launch and run    the SAS program, so you can review the resulting output to convince yourself that the code did indeed generate the desired treatment plan. You should see that five of the subjects were randomly assigned to the A=1, B=1 group, five to the A=1, B=2 group, five to the A=1, B=3 group, and so on.

Then, if you compare the code to the code from the previous example, the only substantial difference you should see is the difference between the two IF statements. As previously mentioned, the IF statement here divides the 30 subjects into 6 treatment groups and (arbitrarily) assigns the levels of factors A and B to the groups:

Example 34.17 Section  

Thus far, our random assignments have not involved dealing with a blocking factor. As you know, it is natural in some experiments to block some of the experimental units together in an attempt to reduce unnecessary variability in your measurements that might otherwise prevent you from making good treatment comparisons. Suppose, for example, that your workload would prevent you from making more than nine experimental measurements in a day. Then, it would be a good idea then to treat the day as a blocking factor. The following program creates a random assignment for 27 subjects in a randomized block design with one factor having three levels.

Again, my apologies about the formatting that makes the program a little more difficult than usual to read. I needed to format it as such so that I could easily capture the image of the program for you.

It's probably going to be best if you first launch and run    the SAS program, so you can first review the contents of the initial exper2 data set:

EXPER2: Definition of Experimental Units

and then the resulting output that contains the desired treatment plan... first in block-treatment order:

Random Assignments for RBD: Sorted in BLOCK-TRT order

and then in block-unit order:

Random Assignments for RBD: Sorted in BLOCK-UNIT order

As you can see, the exper2 data set is created to contain one observation for each of the experimental units (27 subjects here). The variable unit contains an arbitrary label (1, 2, ..., 30) assigned to each of the experimental units. The variable block , which identifies the block number (1, 2, and 3), divides the experimental units into three equal-sized blocks of nine.

Now, to create the random assignment:

• We use the ranuni function to generate a uniform random number between 0 and 1 for each observation.
• Then, within each block, we sort the data in order of the random number.
• Then, we create a counter variable to count the number of observations within each block: for the first observation within each block ("if first.block "), we set the counter ( k ) to 0; otherwise, we increase the counter by 1 for each observation within the block. (For this to work, we must retain k from iteration to iteration).
• Using an IF-THEN-ELSE construct, within each block , assign the first three units in sorted order ( k =0,1,2) to group 1, the second three ( k =3,4,5) to group 2, and the last three ( k =6,7,8) to group 3.

Depending on how the experiment will be conducted, you can print the random assignment in different orders:

• First, the randomization is printed in order of treatment within each block. This will accommodate experiments for which it is natural to perform the treatments in groups on the randomized experimental units.
• Then, the randomization is printed in order of units within the block. This will accommodate experiments for which it is natural to perform the treatments in random order on consecutive experimental units.

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The Definition of Random Assignment According to Psychology

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Emily is a board-certified science editor who has worked with top digital publishing brands like Voices for Biodiversity, Study.com, GoodTherapy, Vox, and Verywell.

Materio / Getty Images

Random assignment refers to the use of chance procedures in psychology experiments to ensure that each participant has the same opportunity to be assigned to any given group in a study to eliminate any potential bias in the experiment at the outset. Participants are randomly assigned to different groups, such as the treatment group versus the control group. In clinical research, randomized clinical trials are known as the gold standard for meaningful results.

Simple random assignment techniques might involve tactics such as flipping a coin, drawing names out of a hat, rolling dice, or assigning random numbers to a list of participants. It is important to note that random assignment differs from random selection .

While random selection refers to how participants are randomly chosen from a target population as representatives of that population, random assignment refers to how those chosen participants are then assigned to experimental groups.

Random Assignment In Research

To determine if changes in one variable will cause changes in another variable, psychologists must perform an experiment. Random assignment is a critical part of the experimental design that helps ensure the reliability of the study outcomes.

Researchers often begin by forming a testable hypothesis predicting that one variable of interest will have some predictable impact on another variable.

The variable that the experimenters will manipulate in the experiment is known as the independent variable , while the variable that they will then measure for different outcomes is known as the dependent variable. While there are different ways to look at relationships between variables, an experiment is the best way to get a clear idea if there is a cause-and-effect relationship between two or more variables.

Once researchers have formulated a hypothesis, conducted background research, and chosen an experimental design, it is time to find participants for their experiment. How exactly do researchers decide who will be part of an experiment? As mentioned previously, this is often accomplished through something known as random selection.

Random Selection

In order to generalize the results of an experiment to a larger group, it is important to choose a sample that is representative of the qualities found in that population. For example, if the total population is 60% female and 40% male, then the sample should reflect those same percentages.

Choosing a representative sample is often accomplished by randomly picking people from the population to be participants in a study. Random selection means that everyone in the group stands an equal chance of being chosen to minimize any bias. Once a pool of participants has been selected, it is time to assign them to groups.

By randomly assigning the participants into groups, the experimenters can be fairly sure that each group will have the same characteristics before the independent variable is applied.

Participants might be randomly assigned to the control group , which does not receive the treatment in question. The control group may receive a placebo or receive the standard treatment. Participants may also be randomly assigned to the experimental group , which receives the treatment of interest. In larger studies, there can be multiple treatment groups for comparison.

There are simple methods of random assignment, like rolling the die. However, there are more complex techniques that involve random number generators to remove any human error.

There can also be random assignment to groups with pre-established rules or parameters. For example, if you want to have an equal number of men and women in each of your study groups, you might separate your sample into two groups (by sex) before randomly assigning each of those groups into the treatment group and control group.

Random assignment is essential because it increases the likelihood that the groups are the same at the outset. With all characteristics being equal between groups, other than the application of the independent variable, any differences found between group outcomes can be more confidently attributed to the effect of the intervention.

Example of Random Assignment

Imagine that a researcher is interested in learning whether or not drinking caffeinated beverages prior to an exam will improve test performance. After randomly selecting a pool of participants, each person is randomly assigned to either the control group or the experimental group.

The participants in the control group consume a placebo drink prior to the exam that does not contain any caffeine. Those in the experimental group, on the other hand, consume a caffeinated beverage before taking the test.

Participants in both groups then take the test, and the researcher compares the results to determine if the caffeinated beverage had any impact on test performance.

A Word From Verywell

Random assignment plays an important role in the psychology research process. Not only does this process help eliminate possible sources of bias, but it also makes it easier to generalize the results of a tested sample of participants to a larger population.

Random assignment helps ensure that members of each group in the experiment are the same, which means that the groups are also likely more representative of what is present in the larger population of interest. Through the use of this technique, psychology researchers are able to study complex phenomena and contribute to our understanding of the human mind and behavior.

Lin Y, Zhu M, Su Z. The pursuit of balance: An overview of covariate-adaptive randomization techniques in clinical trials . Contemp Clin Trials. 2015;45(Pt A):21-25. doi:10.1016/j.cct.2015.07.011

Sullivan L. Random assignment versus random selection . In: The SAGE Glossary of the Social and Behavioral Sciences. SAGE Publications, Inc.; 2009. doi:10.4135/9781412972024.n2108

Alferes VR. Methods of Randomization in Experimental Design . SAGE Publications, Inc.; 2012. doi:10.4135/9781452270012

Nestor PG, Schutt RK. Research Methods in Psychology: Investigating Human Behavior. (2nd Ed.). SAGE Publications, Inc.; 2015.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Statistics Made Easy

Random Selection vs. Random Assignment

Random selection and random assignment  are two techniques in statistics that are commonly used, but are commonly confused.

Random selection  refers to the process of randomly selecting individuals from a population to be involved in a study.

Random assignment  refers to the process of randomly  assigning  the individuals in a study to either a treatment group or a control group.

You can think of random selection as the process you use to “get” the individuals in a study and you can think of random assignment as what you “do” with those individuals once they’re selected to be part of the study.

The Importance of Random Selection and Random Assignment

When a study uses  random selection , it selects individuals from a population using some random process. For example, if some population has 1,000 individuals then we might use a computer to randomly select 100 of those individuals from a database. This means that each individual is equally likely to be selected to be part of the study, which increases the chances that we will obtain a representative sample – a sample that has similar characteristics to the overall population.

By using a representative sample in our study, we’re able to generalize the findings of our study to the population. In statistical terms, this is referred to as having  external validity – it’s valid to externalize our findings to the overall population.

When a study uses  random assignment , it randomly assigns individuals to either a treatment group or a control group. For example, if we have 100 individuals in a study then we might use a random number generator to randomly assign 50 individuals to a control group and 50 individuals to a treatment group.

By using random assignment, we increase the chances that the two groups will have roughly similar characteristics, which means that any difference we observe between the two groups can be attributed to the treatment. This means the study has  internal validity  – it’s valid to attribute any differences between the groups to the treatment itself as opposed to differences between the individuals in the groups.

Examples of Random Selection and Random Assignment

It’s possible for a study to use both random selection and random assignment, or just one of these techniques, or neither technique. A strong study is one that uses both techniques.

The following examples show how a study could use both, one, or neither of these techniques, along with the effects of doing so.

Example 1: Using both Random Selection and Random Assignment

Study:  Researchers want to know whether a new diet leads to more weight loss than a standard diet in a certain community of 10,000 people. They recruit 100 individuals to be in the study by using a computer to randomly select 100 names from a database. Once they have the 100 individuals, they once again use a computer to randomly assign 50 of the individuals to a control group (e.g. stick with their standard diet) and 50 individuals to a treatment group (e.g. follow the new diet). They record the total weight loss of each individual after one month.

Results:  The researchers used random selection to obtain their sample and random assignment when putting individuals in either a treatment or control group. By doing so, they’re able to generalize the findings from the study to the overall population  and  they’re able to attribute any differences in average weight loss between the two groups to the new diet.

Example 2: Using only Random Selection

Study:  Researchers want to know whether a new diet leads to more weight loss than a standard diet in a certain community of 10,000 people. They recruit 100 individuals to be in the study by using a computer to randomly select 100 names from a database. However, they decide to assign individuals to groups based solely on gender. Females are assigned to the control group and males are assigned to the treatment group. They record the total weight loss of each individual after one month.

Results:  The researchers used random selection to obtain their sample, but they did not use random assignment when putting individuals in either a treatment or control group. Instead, they used a specific factor – gender – to decide which group to assign individuals to. By doing this, they’re able to generalize the findings from the study to the overall population but they are  not  able to attribute any differences in average weight loss between the two groups to the new diet. The internal validity of the study has been compromised because the difference in weight loss could actually just be due to gender, rather than the new diet.

Example 3: Using only Random Assignment

Study:  Researchers want to know whether a new diet leads to more weight loss than a standard diet in a certain community of 10,000 people. They recruit 100 males athletes to be in the study. Then, they use a computer program to randomly assign 50 of the male athletes to a control group and 50 to the treatment group. They record the total weight loss of each individual after one month.

Results:  The researchers did not use random selection to obtain their sample since they specifically chose 100 male athletes. Because of this, their sample is not representative of the overall population so their external validity is compromised – they will not be able to generalize the findings from the study to the overall population. However, they did use random assignment, which means they can attribute any difference in weight loss to the new diet.

Example 4: Using Neither Technique

Study:  Researchers want to know whether a new diet leads to more weight loss than a standard diet in a certain community of 10,000 people. They recruit 50 males athletes and 50 female athletes to be in the study. Then, they assign all of the female athletes to the control group and all of the male athletes to the treatment group. They record the total weight loss of each individual after one month.

Results:  The researchers did not use random selection to obtain their sample since they specifically chose 100 athletes. Because of this, their sample is not representative of the overall population so their external validity is compromised – they will not be able to generalize the findings from the study to the overall population. Also, they split individuals into groups based on gender rather than using random assignment, which means their internal validity is also compromised – differences in weight loss might be due to gender rather than the diet.

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Random Assignment – A Simple Introduction with Examples

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Completing a research or thesis paper is more work than most students imagine. For instance, you must conduct experiments before coming up with conclusions. Random assignment, a key methodology in academic research, ensures every participant has an equal chance of being placed in any group within an experiment. In experimental studies, the random assignment of participants is a vital element, which this article will discuss.

Inhaltsverzeichnis

• 1 Random Assignment – In a Nutshell
• 2 Definition: Random assignment
• 3 Importance of random assignment
• 4 Random assignment vs. random sampling
• 5 How to use random assignment
• 6 When random assignment is not used

Random Assignment – In a Nutshell

• Random assignment is where you randomly place research participants into specific groups.
• This method eliminates bias in the results by ensuring that all participants have an equal chance of getting into either group.
• Random assignment is usually used in independent measures or between-group experiment designs.

Definition: Random assignment

Pearson Correlation is a descriptive statistical procedure that describes the measure of linear dependence between two variables. It entails a sample, control group , experimental design , and randomized design. In this statistical procedure, random assignment is used. Random assignment is the random placement of participants into different groups in experimental research.

Importance of random assignment

Random assessment is essential for strengthening the internal validity of experimental research. Internal validity helps make a casual relationship’s conclusions reliable and trustworthy.

In experimental research, researchers isolate independent variables and manipulate them as they assess the impact while managing other variables. To achieve this, an independent variable for diverse member groups is vital. This experimental design is called an independent or between-group design.

Example: Different levels of independent variables

• In a medical study, you can research the impact of nutrient supplements on the immune (nutrient supplements = independent variable, immune = dependent variable)

Three independent participant levels are applicable here:

• Control group (given 0 dosages of iron supplements)
• The experimental group (low dosage)
• The second experimental group (high dosage)

This assignment technique in experiments ensures no bias in the treatment sets at the beginning of the trials. Therefore, if you do not use this technique, you won’t be able to exclude any alternate clarifications for your findings.

In the research experiment above, you can recruit participants randomly by handing out flyers at public spaces like gyms, cafés, and community centers. Then:

• Place the group from cafés in the control group
• Community center group in the low prescription trial group
• Gym group in the high-prescription group

Even with random participant assignment, other extraneous variables may still create bias in experiment results. However, these variations are usually low, hence should not hinder your research. Therefore, using random placement in experiments is highly necessary, especially where it is ethically required or makes sense for your research subject.

Random assignment vs. random sampling

Simple random sampling is a method of choosing the participants for a study. On the other hand, the random assignment involves sorting the participants selected through random sampling. Another difference between random sampling and random assignment is that the former is used in several types of studies, while the latter is only applied in between-subject experimental designs.

Your study researches the impact of technology on productivity in a specific company.

In such a case, you have contact with the entire staff. So, you can assign each employee a quantity and apply a random number generator to pick a specific sample.

For instance, from 500 employees, you can pick 200. So, the full sample is 200.

Random sampling enhances external validity, as it guarantees that the study sample is unbiased, and that an entire population is represented. This way, you can conclude that the results of your studies can be accredited to the autonomous variable.

After determining the full sample, you can break it down into two groups using random assignment. In this case, the groups are:

• The control group (does get access to technology)
• The experimental group (gets access to technology)

Using random assignment assures you that any differences in the productivity results for each group are not biased and will help the company make a decision.

How to use random assignment

Firstly, give each participant a unique number as an identifier. Then, use a specific tool to simplify assigning the participants to the sample groups. Some tools you can use are:

Random member assignment is a prevailing technique for placing participants in specific groups because each person has a fair opportunity of being put in either group.

Random assignment in block experimental designs

In complex experimental designs , you must group your participants into blocks before using the random assignment technique.

You can create participant blocks depending on demographic variables, working hours, or scores. However, the blocks imply that you will require a bigger sample to attain high statistical power.

After grouping the participants in blocks, you can use random assignments inside each block to allocate the members to a specific treatment condition. Doing this will help you examine if quality impacts the result of the treatment.

Depending on their unique characteristics, you can also use blocking in experimental matched designs before matching the participants in each block. Then, you can randomly allot each partaker to one of the treatments in the research and examine the results.

When random assignment is not used

As powerful a tool as it is, random assignment does not apply in all situations. Like the following:

Comparing different groups

When the purpose of your study is to assess the differences between the participants, random member assignment may not work.

If you want to compare teens and the elderly with and without specific health conditions, you must ensure that the participants have specific characteristics. Therefore, you cannot pick them randomly.

In such a study, the medical condition (quality of interest) is the independent variable, and the participants are grouped based on their ages (different levels). Also, all partakers are tried similarly to ensure they have the medical condition, and their outcomes are tested per group level.

No ethical justifiability

Another situation where you cannot use random assignment is if it is ethically not permitted.

If your study involves unhealthy or dangerous behaviors or subjects, such as drug use. Instead of assigning random partakers to sets, you can conduct quasi-experimental research.

When using a quasi-experimental design , you examine the conclusions of pre-existing groups you have no control over, such as existing drug users. While you cannot randomly assign them to groups, you can use variables like their age, years of drug use, or socioeconomic status to group the participants.

What is the definition of random assignment?

It is an experimental research technique that involves randomly placing participants from your samples into different groups. It ensures that every sample member has the same opportunity of being in whichever group (control or experimental group).

When is random assignment applicable?

You can use this placement technique in experiments featuring an independent measures design. It helps ensure that all your sample groups are comparable.

What is the importance of random assignment?

It can help you enhance your study’s validity . This technique also helps ensure that every sample has an equal opportunity of being assigned to a control or trial group.

When should you NOT use random assignment

You should not use this technique if your study focuses on group comparisons or if it is not legally ethical.

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Parametric and Resampling Statistics (cont):

Random sampling and random assignment.

The major assumption behind traditional parametric procedures--more fundamental than normality and homogeneity of variance--is the assumption that we have randomly sampled from some population (usually a normal one). Of course virtually no study you are likely to run will employ true random sampling, but leave that aside for the moment. To see why this assumption is so critical, consider an example in which we draw two samples, calculate the sample means and variances, and use those as estimates of the corresponding population parameters. For example, we might draw a random sample of anorexic girls (potentially) given one treatment, and a random sample of anorexic girls given another treatment, and use our statistical test to draw inferences about the parameters of the corresponding populations from which the girls were randomly sampled. We would probably like to show that our favorite treatment leads to greater weight gain than the competing treatment, and thus the mean of the population of all girls given our favorite treatment is greater than the mean of the other population. But statistically, it makes no sense to say that the sample means are estimates of the corresponding population parameters unless the samples are drawn randomly from that (those) populations(s). (Using the 12 middle school girls in your third period living-arts class is not going to give you a believable estimate of U. S. (let alone world) weights of pre-adolescent girls.) That is why the assumption of random sampling is so critical. In the extreme, if we don't sample randomly, we can't say anything meaningful about the parameters, so why bother? That is part of the argument put forth by the resampling camp.

Of course, those of us who have been involved in statistics for any length of time recognize this assumption, but we rarely give it much thought. We assume that our sample, though not really random, is a pretty good example of what we would have if we had the resources to draw truly random samples, and we go merrily on our way, confident in the belief that the samples we actually have are "good enough" for the purpose. That is where the parametric folks and the resampling folks have a parting of the ways.

The parametric people are not necessarily wrong in thinking that on occasion nonrandom sampling is good enough. If we are measuring something that would not be expected to vary systematically among participants, such as the effect of specific stimulus variations on visual illusions, then a convenience sample may give acceptable results. But keep in mind that any inferences we draw are not statistical inferences, but logical inferences. Without random sampling we cannot make a statistical inference about the mean of a larger population. But on nonstatistical grounds it may make good sense to assume that we have learned something about how people in general process visual information. But using that kind of argument to brush aside some of the criticisms of parametric tests doesn't diminish the fact that the resampling approach legitimately differs in its underlying philosophy.

The resampling approach, and for now I mean the randomization test approach, and not bootstrapping, really looks at the problem differently. In the first place, people in that area don't give a "population" the centrality that we are used to assigning to it in parametric statistics. They don't speak as if they sit around fondly imagining those lovely bell-shaped distributions with numbers streaming out of them, that we often see in introductory textbooks. In fact, they hardly appear to think about populations at all. And they certainly don't think about drawing random samples from those imaginary populations. Those people are as qualified as you could wish as statisticians, but they don't worry too much about estimating parameters, for which you really do need random samples. They just want to know the likelihood of the sample data falling as they did if treatments were equally effective. And for that, they don't absolutely need to think of populations.

In the history of statistics, the procedures with which we are most familiar were developed on the assumption of random sampling. And they were developed with the expectation that we are trying to estimate the corresponding population mean, variance, or whatever. This idea of "estimation" is central to the whole history of traditional statistics--we estimate population means so that we can (hopefully) conclude that they are different and that the treatments have different effects.

But that is not what the randomization test folks are trying to do. They start with the assumption that samples are probably not drawn randomly, and assume that we have no valid basis (or need) for estimating population parameters. This, I think, is the best reason to think of these procedures as nonparametric procedures, though there are other reasons to call them that. But if we can't estimate population parameters, and thus have no legitimate basis for retaining or rejecting a null hypothesis about those parameters, what basis do we have for constructing any statistical test. It turns out that we have legitimate alternative ways for testing our hypothesis, though I'm not sure that we should even be calling it a null hypothesis.

This difference over the role of random sampling is a critical difference between the two approaches. But that is not all. The resampling people, in particular, care greatly about random assignment . The whole approach is based on the idea of random assignment of cases to conditions. That will appear to create problems later on, but take it as part of the underlying rationale. Both groups certainly think that random assignment to conditions is important, primarily because it rules out alternative explanations for any differences that are found. But the resampling camp goes further, and makes it the center point of their analysis. To put it very succinctly, a randomization test works on the logical principle that if cases were randomly assigned to treatments, and if treatments have absolutely no effect on scores, then a particular score is just as likely to have appeared under one condition than under any other. Notice that the principle of random assignment tells us that if the null hypothesis is true, we could validly shuffle the data and expect to get essentially the same results. This is why random assignment is fundamental to the statistical procedure employed.

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Last revised: 03/01/2007 dch

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