Results

Linear Regression

In order to illustrate the new hierarchical regression, we borrow a data set from Andy Field’s popular statistics books and consider album sales (the criterion variable that we wish to predict) along with advertisement budget, attractiveness of the band, and the number of airplays the album received.

For example, let’s predict album sales using bands’ attractiveness, while having accounted for advertisement budget. In other words, let’s assess the extent to which bands’ attractiveness has predictive worth over and above the advertisement budget. To do so, we specify Sales as the dependent variable, and Adverts and Attract as Covariates; then, in the Model submenu, we can tick Adverts to be included in the null model. In the Statistics submenu, we can choose to show the change in R2, as an additional metric for the new model's performance.

Doing so produces the following results:

Model Summary
Model	R	R²	Adjusted R²	RMSE	R² Change	F Change	df1	df2	p
0	0.578	0.335	0.331	65.991	0.335	99.587	1	198	< .001
1	0.643	0.413	0.407	62.129	0.079	26.380	1	197	< .001

Note. Null model includes Adverts

Both metrics indicate that attractiveness has additional predictive worth over and above the advertisement budget alone.

ANOVA
Model		Sum of Squares	df	Mean Square	F	p
0	Regression	433687.683	1	433687.683	99.587	< .001
	Residual	862264.317	198	4354.870
	Total	1.296e +6	199
1	Regression	535517.467	2	267758.734	69.366	< .001
	Residual	760434.533	197	3860.074
	Total	1.296e +6	199

Note. Null model includes Adverts

Coefficients
Model		Unstandardized	Standard Error	Standardized	t	p
0	(Intercept)	134.140	7.537		17.799	< .001
	Adverts	0.096	0.010	0.578	9.979	< .001
1	(Intercept)	26.341	22.155		1.189	0.236
	Adverts	0.092	0.009	0.556	10.150	< .001
	Attract	16.265	3.167	0.281	5.136	< .001

Note that the coefficient estimates are dependent on which model is specified: the regression coefficient for Adverts is different in the null model and the alternative model.