Significance Testing in Multiple Regression

Significance Testing in Multiple Regression

• Significance testing is used to determine if the correlation coefficient (R) is large enough to conclude that there is a linear relationship between variables in the population.
• The null hypothesis (H0) is that there is no linear relationship between X and Y in the population, and the alternative hypothesis (H1) is that there is a linear relationship.
• The correlation coefficient (R) is used to describe the linear relationship between two variables in a sample.
• To test the significance of R, we calculate the probability (likelihood) that the sample was taken from a population where R = 0.
• When the correlation in the population is zero, the sample correlation (R) will be normally distributed around zero.
• We can calculate the probability of obtaining a sample correlation as extreme or more extreme than the one we obtained, assuming that the correlation in the population is zero.
• This probability is used to determine if the sample correlation is significantly different from zero.

Testing the Significance of B

• In addition to testing the significance of R, we can also test the significance of each predictor (b weight) while controlling for all other predictors.
• The significance of the b weights is used to determine if each predictor is significantly associated with the outcome variable, while controlling for the other predictors.
• The same concept of sampling distributions is used to test the significance of the b weights as is used to test the significance of R.

Key Concepts

• R (Rho) is the population correlation coefficient.
• The sampling distribution of the correlation coefficient is used to test the significance of R.
• The null hypothesis (H0) is that there is no linear relationship between X and Y in the population.
• The alternative hypothesis (H1) is that there is a linear relationship between X and Y in the population.
• Significance testing is used to determine if the correlation coefficient (R) is large enough to conclude that there is a linear relationship between variables in the population.

Significant Testing of Correlations and Slopes

• In multiple regression analysis, we look at the association between predictors and the outcome variable, and the variance explained using R-squared.

Correlation and Regression Equation

• Correlation (r) measures the association between two variables in regression analysis.
• The regression equation is a formula that predicts the value of the outcome variable based on the values of the predictor variables.

Significance Testing in Regression

• We test the significance of the correlation coefficient (r) to see if it is significantly different from zero.
• The null hypothesis states that the correlation in the population (ρ) is zero, and the alternative hypothesis states that it is not zero.
• We calculate the probability (p-value) of getting our sample correlation (r) or a more extreme value, assuming that the null hypothesis is true.

Sampling Distribution and P-Values

• We use the sampling distribution of the correlation coefficient to calculate the p-value.
• If the p-value is less than our critical cut-off point (α = 0.05), we reject the null hypothesis and conclude that the correlation is statistically significant.
• A small p-value indicates that it is unlikely to get our sample correlation (or a more extreme value) if the null hypothesis is true.

Interpreting B Weights

• In regression analysis, b weights represent the change in the outcome variable for a one-unit change in the predictor variable, while controlling for other predictors.
• We test the significance of each b weight to see if it is significantly different from zero.
• A significant b weight indicates that the predictor variable is a significant predictor of the outcome variable, even when controlling for other predictors.

Example: Multiple Regression Analysis

• In our example, we looked at the regression analysis of J.R.E.Q and attendance as predictors of exam performance.
• We found that both J.R.E.Q and attendance are significant predictors of exam performance, even when controlling for the correlation between the two predictors.