Hypothesis Tests in Multiple Regression Analysis
Hypothesis Tests in Multiple Regression Analysis
Multiple regression model: [pic]where p represents the total number of variables in the model.
I. Testing for significance of the overall regression model.
Question of interest: Is the regression relation significant? Are one or more of the independent variables in the model useful in explaining variability in Y and/or predicting future values of Y?
Null Hypothesis: The initial assumption is that there is no relation, which is expressed as: H0: β1 = β2 = … = βp-1 =0.
Alternative Hypothesis: At least one of the independent variables IS useful in explaining/predicting Y, expressed as: H1: At least one βi is [pic]0.
Test Statistic: [pic] which is found on any regression printout
Sampling Distribution: Under the null hypothesis the statistic follows an F-distribution with p-1 and n - p degrees of freedom. Reject in the upper tail of this distribution.
Interpreting Results: If we reject H0 we conclude that the relation is significant/does have explanatory or predictive power. If we fail to reject, we conclude that there isn't any evidence of explanatory power, which suggests that there is no point in using this model.
Text Reference: Section 6.5
II. Testing for the Significance/Contribution of a Single Independent Variable in the Model
Question of interest: Suppose we have a significant multiple regression model. In this model, does a single independent variable of interest, say Xj, contribute to explaining/predicting Y? Or, would the model be just as useful without the inclusion of this variable?
Null Hypothesis: The initial assumption is that the variable does not contribute in this model, which is expressed as: H0: βk = 0.
Alternative Hypothesis: The alternative is that the variable does contribute and should remain in the model: H1: βk [pic]0.
Test Statistic: [pic]which is found on any regression printout.
Sampling Distribution: Under the null hypothesis the statistic follows a t-distribution with n - p degrees of freedom. Reject in the upper or lower tail of this distribution.
Interpreting Results: If we reject H0 we conclude that the independent variable Xk does have explanatory or predictive power in our model. Note that conclusion is model-specific, in that it might change if the model included a different set of independent variables. If we fail to reject, we conclude that there isn't any evidence of explanatory power. That suggests that there is no point in having Xk in this model and we should consider dropping it and re-running the regression analysis.
Text Reference: Section 6.6.
III. A General Test for the Value of βj
Question of interest: Does βj equal a specified value of interest, say [pic]? Or, do we have evidence to state that it is not equal to [pic]? (A two-sided test situation is assumed. Make the obvious adjustment for a one-sided test.)
Null Hypothesis: H0: βj = [pic]
Alternative Hypothesis: H1: βj [pic][pic]
Test Statistic: [pic]which is NOT found on the regression printout. You will, however, find bj and s{bj} on the printout.
Sampling Distribution: Under the null hypothesis the statistic follows a t-distribution with n - p degrees of freedom. Reject in the upper or lower tail of this distribution, making the appropriate adjustment for one-sided tests.
Interpreting Results: If we reject H0 we conclude that we have evidence that βj is not equal to the specified [pic] value (we can refute the claim that βj = [pic]). Otherwise, we can't refute the claim.
Text Reference: I don’t think it’s in there. This is an obvious extension of the test for a zero slope presented in Section 6.6.
IV. Testing for the significance/contribution of a subset of independent variables in the regression model. (The “general linear model” test.)
Question of interest: In the multiple regresion model:
[pic] (full model)
does the subset of independent variables Xg, …, Xp-1 contribute to explaining/predicting Y? Or, would be do just as well if these variables were dropped and we reduced the model to
[pic] (reduced model).
Null Hypothesis: The initial assumption is that the subset does not contribute to the model's explanatory power, which is expressed as: H0: βg = … = βp-1 =0.
Alternative Hypothesis: At least one of the independent variables in the subset IS useful in explaining/predicting Y, expressed as: H1: At least one βi is [pic]0, i = g+1 to p-1.
Test Statistic: You need to run two regressions, one for the full model and one for the reduced model as described above. Then calculate:
[pic]
Sampling Distribution: Under the null hypothesis the statistic follows an F-distribution with p - g and n – p degrees of freedom. Reject in the upper tail of this distribution.
Interpreting Results: If we reject H0 we conclude that at least one independent variable in the subset (Xg, …, Xp-1) does have explanatory or predictive power, so we don't reduce the model by dropping out this subset. If we fail to reject, we conclude we have no evidence that inclusion of the subset of independent variables in the model contributes to explanatory power. This suggests that you may as well drop them out and re-run the regression using the reduced model.
Comment: If p - g = 1, i.e. if the subset consists of a single independent variable, then this F-test is equivalent to the two-sided t-test presented in Part II. In fact, t2 = F. You might recall a similar result from simple regression analysis.
Text Reference: Section 2.8. I’m extending the concept to multiple regression.
Example Application
Data Set: Crime Rates for 50 States
Variables: Y = Auto Theft = (duh) a measure of the auto theft rate
Independent Variables (X’s)
Unemployment = % of civilian labor force unemployed in 1991
Police = Police protection per 10,000 population
In-School = Percent of persons 5-17 years old in 1989 enrolled in public elementary and secondary schools
Unemp*Police, the product of Unemployment and Police
Unemp*In-School, the product of Unemployment and In-School
Police*In-School, the product of Police and In*School
Regression Analysis: Full Model
The regression equation is
Auto Theft = 1358 - 306 Unemployment + 25 Police - 7.5 In-School
+ 8.59 Unemp*Police + 1.65 Unemp*In-School - 0.55 Police*In-School
Predictor Coef StDev T P
Constant 1358 4343 0.31 0.756
Unemploy -306.2 448.8 -0.68 0.499
Police 25.4 109.8 0.23 0.818
In-Schoo -7.46 45.76 -0.16 0.871
Unemp*Po 8.592 3.781 2.27 0.028
Unemp*In 1.650 4.690 0.35 0.727
Police*I -0.550 1.103 -0.50 0.621
S = 175.0 R-Sq = 61.6% R-Sq(adj) = 56.3%
Analysis of Variance
Source DF SS MS F P
Regression 6 2114879 352480 11.50 0.000
Residual Error 43 1317461 30639
Total 49 3432340
Regression Analysis: Reduced Model (Interaction terms removed from model)
The regression equation is
Auto Theft = 661 + 48.0 Unemployment + 31.3 Police - 14.4 In-School
Predictor Coef StDev T P
Constant 661.4 610.0 1.08 0.284
Unemploy 48.05 16.96 2.83 0.007
Police 31.291 5.370 5.83 0.000
In-Schoo -14.401 6.128 -2.35 0.023
S = 181.8 R-Sq = 55.7% R-Sq(adj) = 52.8%
Analysis of Variance
Source DF SS MS F P
Regression 3 1912002 637334 19.28 0.000
Residual Error 46 1520338 33051
Total 49 3432340
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