linear regression hypothesis testing calculator

Regression line equation

Reporting linear regression in apa style, line fit plot, prediction online, residual plot, interpretation of the results, regression anova, residuals normality, calculation, linear regression calculator, what is linear regression, what is "ordinary least squares", why do you need linear regression, how to calculate linear regression.

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Linear Regression Calculator

The Linear Regression Calculator uses the following formulas:

Linear Regression Calculator

Instructions: Perform a regression analysis by using the Linear Regression Calculator , where the regression equation will be found and a detailed report of the calculations will be provided, along with a scatter plot. All you have to do is type your X and Y data. Optionally, you can add a title and add the name of the variables.

More about this Linear Regression Calculator

A linear regression model corresponds to a linear model that minimizes the sum of squared errors for a set of pairs \((X_i, Y_i)\).

This is, you assume the existence of a model which in its simplified form is \(Y = \alpha + \beta X\) and then you take note of the discrepancies (errors) found when using this linear model to predict the set of given data.

For each \(X_i\) in the data, you compute \(\hat Y_i = \alpha + \beta X_i\), and you compute the error by measuring \(Y_i - \hat Y_i\). More specifically, in this case you take the square of each discrepancy/error and you add up ALL these square errors.

The objective of a regression calculator is to find the best values of \(\alpha\) and \(\beta\) so that the sum of squared errors is as small as possible.

Regression Formula

The linear regression equation, also known as least squares equation has the following form: \(\hat Y = a + b X\), where the regression coefficients are the values of \(a\) and \(b\).

The question is: How to calculate the regression coefficients? The regression coefficients are computed by this regression calculator as follows:

These are the formulas you used if you were to calculate the regression equation by hand, but likely you will prefer to use a calculator (our regression calculator ) which will show you the important steps.

This linear regression formula is interpreted as follows: The coefficient \(b\) is known as the slope coefficient, and the coefficient \(a\) is known as the y-intercept.

If instead of a linear model, you would like to use a non-linear model, then you should consider instead a polynomial regression calculator , which allows you to use powers of the independent variable.

Linear regression calculator Steps

First of all, you want to assess whether or not it makes sense to run a regression analysis. So then first you should run this correlation coefficient calculator to see if there is a significant degree of linear association between the the variables.

In other words, it only makes sense to run a regression analysis the correlation coefficient is strong enough to make a case for a linear regression model. Also, you should use this scatter plot calculator to ensure that the visual pattern is indeed linear.

It is conceivable that a correlation coefficient is close to 1, but yet the pattern of association is not linear at all.

The steps to conduct a regression analysis are:

Step 1: Get the data for the dependent and independent variable in column format.

Step 2: Type in the data or you can paste it if you already have in Excel format for example.

Step 3: Press "Calculate".

This regression equation calculator with steps will provide you with all the calculations required, in an organized manner, so that you can clearly understand all the steps of the process.

Regression Residuals

How do we assess if a linear regression model is good? You may think "easy, just look at the scatterplot ". In reality, math and statistics tend to go beyond where the eye meets the graph. It is usually risky to rely solely on the scatterplot to assess the quality of the model.

In terms of goodness of fit, one way of assessing the quality of fit of a linear regression model is by computing the coefficient of determination , indicates the proportion of variation that in the dependent variable that is explained by the independent variable.

In linear regression, the fulfillment of the assumptions is crucial so that the estimates of the regression coefficient have good properties (being unbiased, minimum variance, among others).

In order to asses the linear regression assumptions, you will need to take a look at the residuals. For that purpose, you can take a look at our residual calculator .

Predictive Power of a Regression Equation

How can you tell if the regression equation found is good? Or a better question, how to know whether or not the regression equation estimated has good predictive power?

What you need to do is to compute the coefficient of determination , which tells you the amount of variation in the dependent variable that is explained by the dependent variable(s).

For a simple regression model (with one independent variable), the coefficient of determination is simply computed by squaring the correlation coefficient.

For example, if the correlation coefficient is r = 0.8, then the coefficient of determination is \(r^2 = 0.8^2 = 0.64\) and the interpretation is that 64% of the variation in the dependent variable are explained by the independent variable in this model.

Polynomial regression

As we have mentioned before, there are times where the linear regression is simply not appropriate, because there is a clear non-liner pattern governing the relationship between two variables.

Your first signal that polynomial regression should be used instead of linear regression is to see that there is a curvilinear pattern in the data presented by the scatterplot.

If that is the case, you could try this polynomial regression calculator , to estimate a non-linear model that has a better chance of having a better fit.

What is given by this online linear regression calculator?

First, you get a tabulation of the data, and you calculate the corresponding squares and cross-multiplications to get the required sum of squared values, needed to apply the regression formula.

Once that is all neatly shown in a table with all the needed columns, the regression formulas will be shown, with the correct values being plugged in and then with a conclusion about the linear regression model that was estimated from the data.

Also, a scatter plot is constructed in order to assess how tight the linear association is between the variables, which gives an indication of how good the linear regression model is.

Is r2 the regression coefficient?

No. Technically, the regression coefficients are the coefficients estimated that are part of the regression model. The r2 coefficient is called the coefficient of determination.

The coefficient r2 is also computed from sample data, but it is not a regression coefficient, but it does not mean it is not important. The r2 coefficient is important because it gives an estimation of the percentage of variation explained by the model.

How to do linear regression in Excel?

Excel has the ability of conducting linear regression by either directly using the commands "=SLOPE()" and "=INTERCEPT()", or by using the Data Analysis menu.

But Excel does not show all the steps like our regression calculator does.

Other calculators related to linear regression

This regression equation calculator is only one among many calculators of interest when dealing with linear models. You may also be interested in computing the correlation coefficient , or to construct a scatter plot with the data provided.

What is the coefficient of determination?

The coefficient of determination , or R^2 is a measurement of the proportion of the variation in the dependent variable that is explained by the independent variable.

For example, assume that we have a coefficient of determination of R^2 = 0.67 when estimating a linear regression of Y as a function of X, then the interpretation is that X explains 67% of the variation in Y.

What happens when you have more variables

You could potentially have more than one independent variable. For example, you may be interested in estimating Y in terms of two variables X1 and X2. In that case, you need to calculate a multiple linear regression model, where the idea is essentially the same: find the hyperplane that minimize the sum of squared errors.

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Linear Regression Calculator

This simple linear regression calculator uses the least squares method to find the line of best fit for a set of paired data, allowing you to estimate the value of a dependent variable ( Y ) from a given independent variable ( X ).

The line of best fit is described by the equation ŷ = bX + a , where b is the slope of the line and a is the intercept (i.e., the value of Y when X = 0). This calculator will determine the values of b and a for a set of data comprising two variables, and estimate the value of Y for any specified value of X .

To begin, you need to add paired data into the two text boxes immediately below (either one value per line or as a comma delimited list), with your independent variable in the X Values box and your dependent variable in the Y Values box. For example, if you wanted to generate a line of best fit for the association between height and shoe size, allowing you to predict shoe size on the basis of a person's height, then height would be your independent variable and shoe size your dependent variable).

This calculator can estimate the value of a dependent variable ( Y ) for any specified value of an independent variable ( X ). Simply add the X values for which you wish to generate an estimate into the Estimate box below (either one value per line or as a comma delimited list).

Note : If you just want to generate the regression equation that describes the line of best fit, leave the box below blank.

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T test calculator

A t test compares the means of two groups. There are several types of two sample t tests and this calculator focuses on the three most common: unpaired, welch's, and paired t tests. Directions for using the calculator are listed below, along with more information about two sample t tests and help on which is appropriate for your analysis. NOTE: This is not the same as a one sample t test; for that, you need this One sample t test calculator .

1. Choose data entry format

Caution: Changing format will erase your data.

2. Choose a test

Help me choose

3. Enter data

Help me arrange the data

4. View the results

What is a t test.

A t test is used to measure the difference between exactly two means. Its focus is on the same numeric data variable rather than counts or correlations between multiple variables. If you are taking the average of a sample of measurements, t tests are the most commonly used method to evaluate that data. It is particularly useful for small samples of less than 30 observations. For example, you might compare whether systolic blood pressure differs between a control and treated group, between men and women, or any other two groups.

This calculator uses a two-sample t test, which compares two datasets to see if their means are statistically different. That is different from a one sample t test , which compares the mean of your sample to some proposed theoretical value.

The most general formula for a t test is composed of two means (M1 and M2) and the overall standard error (SE) of the two samples:

See our video on How to Perform a Two-sample t test for an intuitive explanation of t tests and an example.

How to use the t test calculator

• Choose your data entry format . This will change how section 3 on the page looks. The first two options are for entering your data points themselves, either manually or by copy & paste. The last two are for entering the means for each group, along with the number of observations (N) and either the standard error of that mean (SEM) or standard deviation of the dataset (SD) standard error. If you have already calculated these summary statistics, the latter options will save you time.
• Choose a test from the three options: Unpaired t test, Welch's unpaired t test, or Paired t test. Use our Ultimate Guide to t tests if you are unsure which is appropriate, as it includes a section on "How do I know which t test to use?". Notice not all options are available if you enter means only.
• Enter data for the test, based on the format you chose in Step 1.
• Click Calculate Now and View the results. All options will perform a two-tailed test .

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Sign up for more information on how to perform t tests and other common statistical analyses.

Common t test confusion

In addition to the number of t test options, t tests are often confused with completely different techniques as well. Here's how to keep them all straight.

Correlation and regression are used to measure how much two factors move together. While t tests are part of regression analysis, they are focused on only one factor by comparing means in different samples.

ANOVA is used for comparing means across three or more total groups. In contrast, t tests compare means between exactly two groups.

Finally, contingency tables compare counts of observations within groups rather than a calculated average. Since t tests compare means of continuous variable between groups, contingency tables use methods such as chi square instead of t tests.

Assumptions of t tests

Because there are several versions of t tests, it's important to check the assumptions to figure out which is best suited for your project. Here are our analysis checklists for unpaired t tests and paired t tests , which are the two most common. These (and the ultimate guide to t tests ) go into detail on the basic assumptions underlying any t test:

• Exactly two groups
• Sample is normally distributed
• Independent observations
• Unequal or equal variance?
• Paired or unpaired data?

Interpreting results

The three different options for t tests have slightly different interpretations, but they all hinge on hypothesis testing and P values. You need to select a significance threshold for your P value (often 0.05) before doing the test.

While P values can be easy to misinterpret , they are the most commonly used method to evaluate whether there is evidence of a difference between the sample of data collected and the null hypothesis. Once you have run the correct t test, look at the resulting P value. If the test result is less than your threshold, you have enough evidence to conclude that the data are significantly different.

If the test result is larger or equal to your threshold, you cannot conclude that there is a difference. However, you cannot conclude that there was definitively no difference either. It's possible that a dataset with more observations would have resulted in a different conclusion.

Depending on the test you run, you may see other statistics that were used to calculate the P value, including the mean difference, t statistic, degrees of freedom, and standard error. The confidence interval and a review of your dataset is given as well on the results page.

Graphing t tests

This calculator does not provide a chart or graph of t tests, however, graphing is an important part of analysis because it can help explain the results of the t test and highlight any potential outliers. See our Prism guide for some graphing tips for both unpaired and paired t tests.

Prism is built for customized, publication quality graphics and charts. For t tests we recommend simply plotting the datapoints themselves and the mean, or an estimation plot . Another popular approach is to use a violin plot, like those available in Prism.

For more information

Our ultimate guide to t tests includes examples, links, and intuitive explanations on the subject. It is quite simply the best place to start if you're looking for more about t tests!

If you enjoyed this calculator, you will love using Prism for analysis. Take a free 30-day trial to do more with your data, such as:

• Clear guidance to pick the right t test and detailed results summaries
• Custom, publication quality t test graphics, violin plots, and more
• More t test options, including normality testing as well as nested and multiple t tests
• Non-parametric test alternatives such as Wilcoxon, Mann-Whitney, and Kolmogorov-Smirnov

Check out our video on how to perform a t test in Prism , for an example from start to finish!

Remember, this page is just for two sample t tests. If you only have one sample, you need to use this calculator instead.

We Recommend:

Analyze, graph and present your scientific work easily with GraphPad Prism. No coding required.

Descriptive Statistics

Hypothesis test, online statistics calculator.

On Statisty you can statistically analyse your data online. Simply copy your own data into the table above and select the variables you want to analyse.

Statisty thus is a free statistical software that makes your calculations directly online. In contrast to SPSS, JASP or Excel, nothing needs to be installed in order to statistically evaluate your data.

Depending on how many variables you click on and what scale level they have, the appropriate tests are calculated.

• One sample t-Test
• Independent t-test
• Paired t-Test
• Binomial Test
• Chi-Square Test
• One-way ANOVA
• Two-way ANOVA
• Repeated measures ANOVA
• Two-way ANOVA with repeated measures
• Mann-Whitney U-test
• Wilcoxon Signed-Rank test
• Kruskal-Wallis Test
• Friedman-Test
• Correlation analysis
• Pearson correlation
• Spearman correlation
• Simple Lineare Regression
• Multiple Lineare Regression
• Logistische Regression

Statistics App

The results are then displayed clearly. First you get the descriptive statistics and then the appropriate hypothesis test. Of course, you can also calculate a linear regression or a logistic regression .

If you like also have a look at the Online Statistics Calculator at DATAtab

For optimal use, please visit DATAtab on your desktop PC!

Dependent Variable:

Independent variable:, regression analysis calculator.

With DATAtab the calculation of a multiple regression analysis goes incredibly easy and directly online! To calculate a regression analysis, simply select a dependent variable and one or more independent variables. Depending on what you have selected, DATAtab automatically calculates:

• Linear Regression
• Multiple Linear Regression
• Logistic Regression

Just try it out with the example data which are already loaded! To calculate a regression analysis, simply select one dependent variable and one or more independent variables.

If you want to use your own data, simply clear the upper table and paste your own data into the Regression Calculator. DATAtab easily calculates online your regression analyses and creates various regression models for you. So just get started, only the three steps are necessary:

• Copy your data into the table of the regression analysis calculator.
• Select a dependent variable.
• Select one or more independent variables.

You can now read from the calculated regression model what the influence of the dependent variables on the independent variable is. This allows you to easily calculate a regression online without SPSS or Excel. It does not matter whether you want to calculate a linear regression online or a logistic regression .

Linear Regression Calculator

Do you want to calculate a linear regression? So you want to get the relationship between one dependent variable and several independent variables. Then select a metric dependent variable and one or more independent variables.

Multiple regression Calculator

Perform a Multiple Linear Regression with our easy, Online Statistical Software. The multiple linear regression calculator uses the least squares method to determine the regression coefficients optimally. The regression coefficients can then be used to interpret how the independent variables affect the dependent variable.

If you have selected more than one independent variable, a multiple linear regression is automatically calculated. In the results you can see how big the influence of the different independent variables is on the dependent variable.

Do you want to know the influence of one variable on another, but want to exclude the influence of other variables? You can do this with a multiple linear regression.

Logistic Regression Calculator

Your goal is to analyze the relationship between a dependent categorical variable and several independent variables in a model? For this you have to calculate a logistic regression! To do so, select a categorical dependent variable and several independent variables.

For example, with a logistic Regression you can find out which variables have an influence on a disease.

Regression analysis calculator

On DATAtab you have the possibility to use the linear regression calculator online . The calculator allows you to model the linear relationship between two or more variables online.

The regression statistics calculator therefore provides you with all relevant statistical values for your data. If you want to calculate the regression line, all you need to do is read the B values in the output table. The result of the regression calculator then looks like this:

Here you will find all the key figures you need, the model summary, the significance test of the whole model and the coefficients. If you need more information about regression analysis, please have a look at our tutorial section:

• Logistic regression
• Linear regression

DATAtab is also available in German, French and Spanish. Visit the Regression Rechner on the German page, the Calculatrice de régression on the French page or the Calculadora de regresión on the Spanish page.

Cite DATAtab: DATAtab Team (2024). DATAtab: Online Statistics Calculator. DATAtab e.U. Graz, Austria. URL https://datatab.net

Teach yourself statistics

Hypothesis Test for Regression Slope

This lesson describes how to conduct a hypothesis test to determine whether there is a significant linear relationship between an independent variable X and a dependent variable Y .

The test focuses on the slope of the regression line

Y = Β 0 + Β 1 X

where Β 0 is a constant, Β 1 is the slope (also called the regression coefficient), X is the value of the independent variable, and Y is the value of the dependent variable.

If we find that the slope of the regression line is significantly different from zero, we will conclude that there is a significant relationship between the independent and dependent variables.

Test Requirements

The approach described in this lesson is valid whenever the standard requirements for simple linear regression are met.

• The dependent variable Y has a linear relationship to the independent variable X .
• For each value of X, the probability distribution of Y has the same standard deviation σ.
• The Y values are independent.
• The Y values are roughly normally distributed (i.e., symmetric and unimodal ). A little skewness is ok if the sample size is large.

The test procedure consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

State the Hypotheses

If there is a significant linear relationship between the independent variable X and the dependent variable Y , the slope will not equal zero.

H o : Β 1 = 0

H a : Β 1 ≠ 0

The null hypothesis states that the slope is equal to zero, and the alternative hypothesis states that the slope is not equal to zero.

Formulate an Analysis Plan

The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the following elements.

• Significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use a linear regression t-test (described in the next section) to determine whether the slope of the regression line differs significantly from zero.

Analyze Sample Data

Using sample data, find the standard error of the slope, the slope of the regression line, the degrees of freedom, the test statistic, and the P-value associated with the test statistic. The approach described in this section is illustrated in the sample problem at the end of this lesson.

SE = s b 1 = sqrt [ Σ(y i - ŷ i ) 2 / (n - 2) ] / sqrt [ Σ(x i - x ) 2 ]

• Slope. Like the standard error, the slope of the regression line will be provided by most statistics software packages. In the hypothetical output above, the slope is equal to 35.

t = b 1 / SE

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

Interpret Results

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

Test Your Understanding

The local utility company surveys 101 randomly selected customers. For each survey participant, the company collects the following: annual electric bill (in dollars) and home size (in square feet). Output from a regression analysis appears below.

Is there a significant linear relationship between annual bill and home size? Use a 0.05 level of significance.

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

H o : The slope of the regression line is equal to zero.

H a : The slope of the regression line is not equal to zero.

• Formulate an analysis plan . For this analysis, the significance level is 0.05. Using sample data, we will conduct a linear regression t-test to determine whether the slope of the regression line differs significantly from zero.

We get the slope (b 1 ) and the standard error (SE) from the regression output.

b 1 = 0.55       SE = 0.24

We compute the degrees of freedom and the t statistic, using the following equations.

DF = n - 2 = 101 - 2 = 99

t = b 1 /SE = 0.55/0.24 = 2.29

where DF is the degrees of freedom, n is the number of observations in the sample, b 1 is the slope of the regression line, and SE is the standard error of the slope.

• Interpret results . Since the P-value (0.0242) is less than the significance level (0.05), we cannot accept the null hypothesis.

Linear Regression Calculator

This calculator produces a linear regression equation based on values for a predictor variable and a response variable.

Simply enter a list of values for a predictor variable and a response variable in the boxes below, then click the “Calculate” button:

Predictor values:

Response values:

Linear Regression Equation:

ŷ = 0.9694 + ( 7.7673 )*x

Goodness of Fit:

R Square: 0.8282

Interpretation:

When the predictor variable is equal to 0, the average value for the response variable is 0.9694 .

Each one unit increase in the predictor variable is associated with an average change of ( 7.7673 ) in the response variable.

82.82 % of the variation in the response variable can be explained by the predictor variable.

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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Simple Linear Regression Calculator

• Values of the response variable $y$ vary according to a normal distribution with standard deviation $\sigma$ for any value of the explanatory variable $x$. The quantity $\sigma$ is an unknown parameter.
• Repeated values of $y$ are independent of one another.
• The relationship between the mean response of $y$ (denoted as $\mu_y$) and explanatory variable $x$ is a straight line given by $\mu_y=\alpha+\beta x$ where $\alpha$ and $\beta$ are unknown parameters.

Multiple Regression Calculator

The difference between a multiple regression and a simple linear regression is that in a multiple regression there are more than one independent variable (x). Although it's not stated in its name, there is still a linear relationship between the dependent (y) and independent variables in multiple regression. Generally speaking, there are a total of $p$ independent variables in multiple regression. Here, $p$ can take any value greater than one. If $p$ is equal to one, then it is just a simple linear regression. The estimated multiple regression equation is given below.

As in simple linear regression, the coefficient in multiple regression are found using the least squared method. That is, the coefficients are chosen such that the sum of the square of the residuals are minimized. The difference is that the formulas for the coefficients can be expressed using algebra in simple linear regression. In multiple regression, the formulas for the coefficients require the use of more advanced math, specifically matrix algebra. Because of this, calculation by hand of coefficients in multiple regression is usually avoided and the focus is on the interpretation of the coefficients.

The coefficient of determination, or r-squared, in multiple regression is computed in the same way as it is in simple linear regression. However, there is a problem in using it in multiple regression. That problem is that the r-squared naturally increases as you add more independent variables, even if those independent variables aren't relevant. To solve this problem, the adjusted coefficient of determination is preferred in multiple regression. The formula for the adjusted r-squared is given below.

The interpretation of the coefficient of determination is the same as it is in simple linear regression. That is, the first step is to convert it from a decimal to a percentage by multiplying by 100%. Then its the percentage of variability in the dependent variable explained by the estimated regression equation. In the case of multiple regression, you'd want to use the adjusted r-squared instead of the regular r-squared. The interpretation of the slope coefficients are that they give the predicted changed in the dependent variable corresponding to a one-unit increase in the independent variable, holding the other independent variables constant.

As in simple linear regression, testing for significance for multiple regression involves either the use of the F-test or t-test. However, while the two tests are the same in simple regression, they are different in multiple regression. In multiple regression, the F-test is a simultaneous test for significance for all the independent variables. If the null hypothesis in the F-test is rejected, then at least one of the independent variables is significant. If the null hypothesis is not rejected, then none of the independent variables are significant.

If the F test passes (i.e., null hypothesis is rejected) in multiple regression, then we can proceed to do t tests. Once we know that at least one of the independent variables are significant, the t-tests can be used to determine which ones are significant. So the t-tests are performed on each individual independent variable. Rejecting the null hypothesis in a t-test means that the independent variable is significant. So while the two tests of significance are subsitutes in simple regression, they complement each other in multiple regression.

One of the obstacles commonly reached in multiple regression is running into categorical (or qualitative) data. Categorical data is data that does not involve numbers, such as gender or country. The problem with using categorical data in regression is that the least squares method requires numerical data to compute the estimated coefficients. This issues is resolved in multiple regression through the use of dummy variables. A dummy variable takes the value one for one category and zero for the other category. When there are more than two categories, more than one dummy variable is used.

While a multiple regression can provide great predictive power, oftentimes a simple linear regression is enough. To compute a simple linear regression and the associated statistics, visit the Simple Regression Calculator . The F test and t test in multiple regression are two examples of hypothesis tests. To perform hypothesis tests, visit the Hypothesis Testing Calculator .

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Linear regression hypothesis testing: Concepts, Examples

In relation to machine learning , linear regression is defined as a predictive modeling technique that allows us to build a model which can help predict continuous response variables as a function of a linear combination of explanatory or predictor variables. While training linear regression models, we need to rely on hypothesis testing in relation to determining the relationship between the response and predictor variables. In the case of the linear regression model, two types of hypothesis testing are done. They are T-tests and F-tests . In other words, there are two types of statistics that are used to assess whether linear regression models exist representing response and predictor variables. They are t-statistics and f-statistics. As data scientists , it is of utmost importance to determine if linear regression is the correct choice of model for our particular problem and this can be done by performing hypothesis testing related to linear regression response and predictor variables. Many times, it is found that these concepts are not very clear with a lot many data scientists. In this blog post, we will discuss linear regression and hypothesis testing related to t-statistics and f-statistics . We will also provide an example to help illustrate how these concepts work.

Table of Contents

What are linear regression models?

A linear regression model can be defined as the function approximation that represents a continuous response variable as a function of one or more predictor variables. While building a linear regression model, the goal is to identify a linear equation that best predicts or models the relationship between the response or dependent variable and one or more predictor or independent variables.

There are two different kinds of linear regression models. They are as follows:

• Simple or Univariate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and one predictor or independent variable. The form of the equation that represents a simple linear regression model is Y=mX+b, where m is the coefficients of the predictor variable and b is bias. When considering the linear regression line, m represents the slope and b represents the intercept.
• Multiple or Multi-variate linear regression models : These are linear regression models that are used to build a linear relationship between one response or dependent variable and more than one predictor or independent variable. The form of the equation that represents a multiple linear regression model is Y=b0+b1X1+ b2X2 + … + bnXn, where bi represents the coefficients of the ith predictor variable. In this type of linear regression model, each predictor variable has its own coefficient that is used to calculate the predicted value of the response variable.

While training linear regression models, the requirement is to determine the coefficients which can result in the best-fitted linear regression line. The learning algorithm used to find the most appropriate coefficients is known as least squares regression . In the least-squares regression method, the coefficients are calculated using the least-squares error function. The main objective of this method is to minimize or reduce the sum of squared residuals between actual and predicted response values. The sum of squared residuals is also called the residual sum of squares (RSS). The outcome of executing the least-squares regression method is coefficients that minimize the linear regression cost function .

The residual e of the ith observation is represented as the following where [latex]Y_i[/latex] is the ith observation and [latex]\hat{Y_i}[/latex] is the prediction for ith observation or the value of response variable for ith observation.

[latex]e_i = Y_i – \hat{Y_i}[/latex]

The residual sum of squares can be represented as the following:

[latex]RSS = e_1^2 + e_2^2 + e_3^2 + … + e_n^2[/latex]

The least-squares method represents the algorithm that minimizes the above term, RSS.

Once the coefficients are determined, can it be claimed that these coefficients are the most appropriate ones for linear regression? The answer is no. After all, the coefficients are only the estimates and thus, there will be standard errors associated with each of the coefficients.  Recall that the standard error is used to calculate the confidence interval in which the mean value of the population parameter would exist. In other words, it represents the error of estimating a population parameter based on the sample data. The value of the standard error is calculated as the standard deviation of the sample divided by the square root of the sample size. The formula below represents the standard error of a mean.

[latex]SE(\mu) = \frac{\sigma}{\sqrt(N)}[/latex]

Thus, without analyzing aspects such as the standard error associated with the coefficients, it cannot be claimed that the linear regression coefficients are the most suitable ones without performing hypothesis testing. This is where hypothesis testing is needed . Before we get into why we need hypothesis testing with the linear regression model, let’s briefly learn about what is hypothesis testing?

Train a Multiple Linear Regression Model using R

Before getting into understanding the hypothesis testing concepts in relation to the linear regression model, let’s train a multi-variate or multiple linear regression model and print the summary output of the model which will be referred to, in the next section. 

The data used for creating a multi-linear regression model is BostonHousing which can be loaded in RStudioby installing mlbench package. The code is shown below:

install.packages(“mlbench”) library(mlbench) data(“BostonHousing”)

Once the data is loaded, the code shown below can be used to create the linear regression model.

attach(BostonHousing) BostonHousing.lm <- lm(log(medv) ~ crim + chas + rad + lstat) summary(BostonHousing.lm)

Executing the above command will result in the creation of a linear regression model with the response variable as medv and predictor variables as crim, chas, rad, and lstat. The following represents the details related to the response and predictor variables:

• log(medv) : Log of the median value of owner-occupied homes in USD 1000’s
• crim : Per capita crime rate by town
• chas : Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
• rad : Index of accessibility to radial highways
• lstat : Percentage of the lower status of the population

The following will be the output of the summary command that prints the details relating to the model including hypothesis testing details for coefficients (t-statistics) and the model as a whole (f-statistics) 

Hypothesis tests & Linear Regression Models

Hypothesis tests are the statistical procedure that is used to test a claim or assumption about the underlying distribution of a population based on the sample data. Here are key steps of doing hypothesis tests with linear regression models:

• Hypothesis formulation for T-tests: In the case of linear regression, the claim is made that there exists a relationship between response and predictor variables, and the claim is represented using the non-zero value of coefficients of predictor variables in the linear equation or regression model. This is formulated as an alternate hypothesis. Thus, the null hypothesis is set that there is no relationship between response and the predictor variables . Hence, the coefficients related to each of the predictor variables is equal to zero (0). So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis for each test states that a1 = 0, a2 = 0, a3 = 0 etc. For all the predictor variables, individual hypothesis testing is done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. Thus, if there are, say, 5 features, there will be five hypothesis tests and each will have an associated null and alternate hypothesis.
• Hypothesis formulation for F-test : In addition, there is a hypothesis test done around the claim that there is a linear regression model representing the response variable and all the predictor variables. The null hypothesis is that the linear regression model does not exist . This essentially means that the value of all the coefficients is equal to zero. So, if the linear regression model is Y = a0 + a1x1 + a2x2 + a3x3, then the null hypothesis states that a1 = a2 = a3 = 0.
• F-statistics for testing hypothesis for linear regression model : F-test is used to test the null hypothesis that a linear regression model does not exist, representing the relationship between the response variable y and the predictor variables x1, x2, x3, x4 and x5. The null hypothesis can also be represented as x1 = x2 = x3 = x4 = x5 = 0. F-statistics is calculated as a function of sum of squares residuals for restricted regression (representing linear regression model with only intercept or bias and all the values of coefficients as zero) and sum of squares residuals for unrestricted regression (representing linear regression model). In the above diagram, note the value of f-statistics as 15.66 against the degrees of freedom as 5 and 194. 
• Evaluate t-statistics against the critical value/region : After calculating the value of t-statistics for each coefficient, it is now time to make a decision about whether to accept or reject the null hypothesis. In order for this decision to be made, one needs to set a significance level, which is also known as the alpha level. The significance level of 0.05 is usually set for rejecting the null hypothesis or otherwise. If the value of t-statistics fall in the critical region, the null hypothesis is rejected. Or, if the p-value comes out to be less than 0.05, the null hypothesis is rejected.
• Evaluate f-statistics against the critical value/region : The value of F-statistics and the p-value is evaluated for testing the null hypothesis that the linear regression model representing response and predictor variables does not exist. If the value of f-statistics is more than the critical value at the level of significance as 0.05, the null hypothesis is rejected. This means that the linear model exists with at least one valid coefficients. 
• Draw conclusions : The final step of hypothesis testing is to draw a conclusion by interpreting the results in terms of the original claim or hypothesis. If the null hypothesis of one or more predictor variables is rejected, it represents the fact that the relationship between the response and the predictor variable is not statistically significant based on the evidence or the sample data we used for training the model. Similarly, if the f-statistics value lies in the critical region and the value of the p-value is less than the alpha value usually set as 0.05, one can say that there exists a linear regression model.

Why hypothesis tests for linear regression models?

The reasons why we need to do hypothesis tests in case of a linear regression model are following:

• By creating the model, we are establishing a new truth (claims) about the relationship between response or dependent variable with one or more predictor or independent variables. In order to justify the truth, there are needed one or more tests. These tests can be termed as an act of testing the claim (or new truth) or in other words, hypothesis tests.
• One kind of test is required to test the relationship between response and each of the predictor variables (hence, T-tests)
• Another kind of test is required to test the linear regression model representation as a whole. This is called F-test.

While training linear regression models, hypothesis testing is done to determine whether the relationship between the response and each of the predictor variables is statistically significant or otherwise. The coefficients related to each of the predictor variables is determined. Then, individual hypothesis tests are done to determine whether the relationship between response and that particular predictor variable is statistically significant based on the sample data used for training the model. If at least one of the null hypotheses is rejected, it represents the fact that there exists no relationship between response and that particular predictor variable. T-statistics is used for performing the hypothesis testing because the standard deviation of the sampling distribution is unknown. The value of t-statistics is compared with the critical value from the t-distribution table in order to make a decision about whether to accept or reject the null hypothesis regarding the relationship between the response and predictor variables. If the value falls in the critical region, then the null hypothesis is rejected which means that there is no relationship between response and that predictor variable. In addition to T-tests, F-test is performed to test the null hypothesis that the linear regression model does not exist and that the value of all the coefficients is zero (0). Learn more about the linear regression and t-test in this blog – Linear regression t-test: formula, example .

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Ajitesh Kumar

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Very informative

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