Multiple Regression Cont.

3/28/2002

This lab follows the completion of the rest of the lecture from Tuesday. It is illustrates the role of multiple variables in accounting for variance, and builds on many of the concepts in Tuesday's class. I would have liked to give you a lab on interaction terms, but we are running out of time. However, you can find a good example at Multreg2, using an example to which I have already alluded. That page contains the worked solution, so you can go through it on your own, even without doing the analyses.

The following question comes from a paper by Guber (1999) published in the Journal of Statistics Education, which is an online journal. (You can get access to the summary by going to http://www.amstat.org/publications/jse/v7n2_abstracts.html .) Interestingly, Deborah Guber is in the Political Science department at UVM, though we have never met.

Guber was interested in the relationship between how much states spend on education and the performance of their students on SAT tests. This is certainly a relevant question for psychologists, so if fits nicely with this course. That simple question can be misleading, as you will see, so she added the percentage of students taking the SAT for each state, and other variables not shown here. Her data are reproduced in abbreviated form below, where SAT refers to the combined Math and Verbal score. The data can be found at SAT.sav.

• Plot a histogram of the individual variables. How would you describe these variables?
  
• Create a scatterplot of the relationship between Expenditure and SAT scores.
  
• Draw a scatterplot of the relationship between SAT scores and the percentage of students taking the SAT (PctSAT).
  
  • Can you explain why we might find such a result?
    
• Run the simple regression of SAT as a function of Expenditure, and then rerun it adding PctSAT. 
  
  • Why would this be called a hierarchical regression?
    
• What does the answer in the previous regression contribute to our understanding of what is going on?
  
• Rerun the last multiple regression, but this time calculate the residuals and plot  them as a function of the predicted value. Does this leave you somewhat concerned?
  
• Create a new variable which is the square root of PctSAT and use this with PctSAT and Expenditure to predict SAT. What does this show? What happens to a plot of the residuals against the predicted value?? (This was a suggestion from Guber, and a good one at that.)

 

State	Expend	Pct	Combined		State		Expend	Pct	Combined
"Alabama"	4.405	8	1029		"Nebraska"		5.935	9	1050
"Alaska"	8.963	47	934		"Nevada"		5.160	30	917
"Arizona"	4.778	27	944		"New Hamp"		5.859	70	935
"Arkansas"	4.459	6	1005		"New Jersey"		9.774	70	898
"Calif"	4.992	45	902		"New Mexico"		4.586	11	1015
"Colorado"	5.443	29	980		"New York"		9.623	74	892
"Conn"	8.817	81	908		"North Carol”		.	60	865
"Delaware"	7.030	68	897		"North Dak"		4.775	5	1107
"Florida"	5.718	48	889		"Ohio"		6.162	23	975
"Georgia"	5.193	65	854		"Oklahoma"		4.845	9	1027
"Hawaii"	6.078	57	889		"Oregon"		6.436	51	947
"Idaho"	4.210	15	979		"Pennsylvania"		7.109	70	880
"Illinois"	6.136	13	1048		"Rhode Island"		7.469	70	888
"Indiana"	5.826	58	882		"South Carol”		.	58	844
"Iowa"	5.483	5	1099		"South Dak"		4.775	5	1068
"Kansas"	5.817	9	1060		"Tennessee"		4.388	12	1040
"Kentucky"	5.217	11	999		"Texas"		5.222	47	893
"Louisiana"	4.761	9	1021		"Utah"		3.656	4	1076
"Maine"	6.428	68	896		"Vermont"		6.750	68	901
"Maryland"	7.245	64	909		"Virginia"		5.327	65	896
"Mass"	7.287	80	907		"Wash"		5.906	48	937
"Michigan"	6.994	11	1033			"West Vir"	6.107	17	932
"Minn"	6.000	9	1085			"Wisconsin"	6.930	9	1073
"Miss"	4.080	4	1036			"Wyoming"	6.160	10	1001
"Missouri"	5.383	9	1045
"Montana"	5.692	21	1009

Last revised:  03/25/02