Simple and Multiple Regression
Chapter Outline1.0 Introduction
1.1 A First Regression Analysis
1.2 Examining Data
1.3 Simple linear regression
1.4 Multiple regression
1.5 Transforming variables
1.6 Summary
1.7 For more information
1.0 Introduction
This web book is composed of four chapters covering a variety of topics about using SAS for regression. We should emphasize that this book is about "data analysis" and that it demonstrates how SAS can be used for regression analysis, as opposed to a book that covers the statistical basis of multiple regression. We assume that you have had at least one statistics course covering regression analysis and that you have a regression book that you can use as a reference (see the Regression With SAS page and our Statistics Books for Loan page for recommended regression analysis books). This book is designed to apply your knowledge of regression, combine it with instruction on SAS, to perform, understand and interpret regression analyses.
This first chapter will cover topics in simple and multiple regression, as well as the supporting tasks that are important in preparing to analyze your data, e.g., data checking, getting familiar with your data file, and examining the distribution of your variables. We will illustrate the basics of simple and multiple regression and demonstrate the importance of inspecting, checking and verifying your data before accepting the results of your analysis. In general, we hope to show that the results of your regression analysis can be misleading without further probing of your data, which could reveal relationships that a casual analysis could overlook.
In this chapter, and in subsequent chapters, we will be using a data file that was created by randomly sampling 400 elementary schools from the California Department of Education's API 2000 dataset. This data file contains a measure of school academic performance as well as other attributes of the elementary schools, such as, class size, enrollment, poverty, etc.
You can access this data file over the web by clicking on elemapi.sas7bdat, or by visiting the Regression with SAS page where you can download all of the data files used in all of the chapters of this book. The examples will assume you have stored your files in a folder called c:\sasreg, but actually you can store the files in any folder you choose, but if you run these examples be sure to change c:\sasreg\ to the name of the folder you have selected.
1.1 A First Regression Analysis
Let's dive right in and perform a regression analysis using the variables api00, acs_k3, meals and full. These measure the academic performance of the school (api00), the average class size in kindergarten through 3rd grade (acs_k3), the percentage of students receiving free meals (meals) - which is an indicator of poverty, and the percentage of teachers who have full teaching credentials (full). We expect that better academic performance would be associated with lower class size, fewer students receiving free meals, and a higher percentage of teachers having full teaching credentials. Below, we use proc reg for running this regression model followed by the SAS output.
proc reg data="c:\sasreg\elemapi";
model api00 = acs_k3 meals full;
run;
The REG Procedure
Model: MODEL1
Dependent Variable: api00 api 2000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 2634884 878295 213.41 <.0001
Error 309 1271713 4115.57673
Corrected Total 312 3906597
Root MSE 64.15276 R-Square 0.6745
Dependent Mean 596.40575 Adj R-Sq 0.6713
Coeff Var 10.75656
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 906.73916 28.26505 32.08 <.0001
acs_k3 avg class size k-3 1 -2.68151 1.39399 -1.92 0.0553
meals pct free meals 1 -3.70242 0.15403 -24.04 <.0001
full pct full credential 1 0.10861 0.09072 1.20 0.2321
Should we take these results and write them up for publication? From these results, we would conclude that lower class sizes are related to higher performance, that fewer students receiving free meals is associated with higher performance, and that the percentage of teachers with full credentials was not related to academic performance in the schools. Before we write this up for publication, we should do a number of checks to make sure we can firmly stand behind these results. We start by getting more familiar with the data file, doing preliminary data checking, looking for errors in the data.
1.2 Examining data
First, let's use proc contents to learn more about this data file. We can verify how many observations it has and see the names of the variables it contains.
proc contents data="c:\sasreg\elemapi" ; run;
The CONTENTS Procedure
Data Set Name: c:\sasreg\elemapi Observations: 400
Member Type: DATA Variables: 21
Engine: V8 Indexes: 0
Created: 4:58 Saturday, January 9, 1960 Observation Length: 83
Last Modified: 4:58 Saturday, January 9, 1960 Deleted Observations: 0
Protection: Compressed: NO
Data Set Type: Sorted: NO
Label:
-----Engine/Host Dependent Information-----
Data Set Page Size: 8192
Number of Data Set Pages: 5
First Data Page: 1
Max Obs per Page: 98
Obs in First Data Page: 56
Number of Data Set Repairs: 0
File Name: c:\sasreg\elemapi.sas7bdat
Release Created: 7.0000M0
Host Created: WIN_NT
-----Alphabetic List of Variables and Attributes-----
# Variable Type Len Pos Label
-----------------------------------------------------------------------------
11 acs_46 Num 3 39 avg class size 4-6
10 acs_k3 Num 3 36 avg class size k-3
3 api00 Num 4 12 api 2000
4 api99 Num 4 16 api 1999
17 avg_ed Num 8 57 avg parent ed
15 col_grad Num 3 51 parent college grad
2 dnum Num 4 8 district number
7 ell Num 3 27 english language learners
19 emer Num 3 73 pct emer credential
20 enroll Num 4 76 number of students
18 full Num 8 65 pct full credential
16 grad_sch Num 3 54 parent grad school
5 growth Num 4 20 growth 1999 to 2000
13 hsg Num 3 45 parent hsg
21 mealcat Num 3 80 Percentage free meals in 3 categories
6 meals Num 3 24 pct free meals
9 mobility Num 3 33 pct 1st year in school
12 not_hsg Num 3 42 parent not hsg
1 snum Num 8 0 school number
14 some_col Num 3 48 parent some college
8 yr_rnd Num 3 30 year round school
We will not go into all of the details of this output. Note that there are 400
observations and 21 variables. We have variables about academic performance in 2000
and 1999 and the change in performance, api00, api99 and growth
respectively. We also have various characteristics of the schools, e.g., class size,
parents education, percent of teachers with full and emergency credentials, and number of
students. Note that when we did our original regression analysis it said that there
were 313 observations, but the proc contents output indicates that we have 400
observations in the data file.If you want to learn more about the data file, you could use proc print to show some of the observations. For example, below we proc print to show the first five observations.
proc print data="c:\sasreg\elemapi"(obs=5) ; run;
m s c g
o n o o r m
g y b a a o m l a a e e
a a r m r i c c t e _ d v n a
s d p p o e _ l s s _ _ g _ g f e r l
O n n i i w a e r i _ _ h h c r s _ u m o c
b u u 0 9 t l l n t k 4 s s o a c e l e l a
s m m 0 9 h s l d y 3 6 g g l d h d l r l t
1 906 41 693 600 93 67 9 0 11 16 22 0 0 0 0 0 . 76 24 247 2
2 889 41 570 501 69 92 21 0 33 15 32 0 0 0 0 0 . 79 19 463 3
3 887 41 546 472 74 97 29 0 36 17 25 0 0 0 0 0 . 68 29 395 3
4 876 41 571 487 84 90 27 0 27 20 30 36 45 9 9 0 1.91000 87 11 418 3
5 888 41 478 425 53 89 30 0 44 18 31 50 50 0 0 0 1.50000 87 13 520 3
This takes up lots of space on the page, but does not give us a lot of
information. Listing our data can be very helpful, but it is more helpful if you list
just the variables you are interested in. Let's list the first 10
observations for the variables that we looked at in our first regression analysis.
proc print data="c:\sasreg\elemapi"(obs=10) ; var api00 acs_k3 meals full; run;
Obs api00 acs_k3 meals full 1 693 16 67 76 2 570 15 92 79 3 546 17 97 68 4 571 20 90 87 5 478 18 89 87 6 858 20 . 100 7 918 19 . 100 8 831 20 . 96 9 860 20 . 100 10 737 21 29 96We see that among the first 10 observations, we have four missing values for meals. It is likely that the missing data for meals had something to do with the fact that the number of observations in our first regression analysis was 313 and not 400.
Another useful tool for learning about your variables is proc means. Below we use proc means to learn more about the variables api00, acs_k3, meals, and full.
proc means data="c:\sasreg\elemapi"; var api00 acs_k3 meals full; run;
The MEANS Procedure Variable Label N Mean Std Dev Minimum ---------------------------------------------------------------------------- api00 api 2000 400 647.6225000 142.2489610 369.0000000 acs_k3 avg class size k-3 398 18.5477387 5.0049328 -21.0000000 meals pct free meals 315 71.9936508 24.3855697 6.0000000 full pct full credential 400 66.0568000 40.2979258 0.4200000 ---------------------------------------------------------------------------- Variable Label Maximum ------------------------------------------- api00 api 2000 940.0000000 acs_k3 avg class size k-3 25.0000000 meals pct free meals 100.0000000 full pct full credential 100.0000000 -------------------------------------------We see that the api00 scores don't have any missing values (because the N is 400) and the scores range from 369-940. This makes sense since the api scores can range from 200 to 1000. We see that the average class size (acs_k3) had 398 valid values ranging from -21 to 25 and 2 are missing. It seems odd for a class size to be -21. The percent receiving free meals (meals) ranges from 6 to 100, but there are only 315 valid values (85 are missing). This seems like a large number of missing values. The percent with full credentials (full) ranges from .42 to 100 with no missing.
We can also use proc freq to learn more about any categorical variables, such as yr_rnd, as shown below.
proc freq data="c:\sasreg\elemapi";
tables yr_rnd;
run;
year round school
Cumulative Cumulative
yr_rnd Frequency Percent Frequency Percent
-----------------------------------------------------------
0 308 77.00 308 77.00
1 92 23.00 400 100.00
The variable yr_rnd is coded 0=No (not year round) and 1=Yes (year
round). Of the 400 schools, 308 are non-year round and 92 are year round,
and none are missing.The above commands have uncovered a number of peculiarities worthy of further examination. For example, let us look further into the average class size by getting more detailed summary statistics for acs_k3 using proc univariate.
proc univariate data="c:\sasreg\elemapi"; var acs_k3; run;
The UNIVARIATE Procedure
Variable: acs_k3 (avg class size k-3)
Moments
N 398 Sum Weights 398
Mean 18.5477387 Sum Observations 7382
Std Deviation 5.00493282 Variance 25.0493526
Skewness -7.1055928 Kurtosis 53.0136683
Uncorrected SS 146864 Corrected SS 9944.59296
Coeff Variation 26.9840594 Std Error Mean 0.25087461
Basic Statistical Measures
Location Variability
Mean 18.54774 Std Deviation 5.00493
Median 19.00000 Variance 25.04935
Mode 19.00000 Range 46.00000
Interquartile Range 2.00000
Tests for Location: Mu0=0
Test -Statistic- -----p Value------
Student's t t 73.93231 Pr > |t| <.0001
Sign M 193 Pr >= |M| <.0001
Signed Rank S 37839 Pr >= |S| <.0001
Quantiles (Definition 5)
Quantile Estimate
100% Max 25
99% 23
95% 21
90% 21
75% Q3 20
50% Median 19
25% Q1 18
10% 17
5% 16
1% -20
0% Min -21
Extreme Observations
----Lowest---- ----Highest---
Value Obs Value Obs
-21 43 22 365
-21 42 23 36
-21 41 23 79
-20 40 23 361
-20 38 25 274
Missing Values
-----Percent Of-----
Missing Missing
Value Count All Obs Obs
. 2 0.50 100.00
Looking in the section labeled Extreme Observations, we see some of the class
sizes are -21 and -20, so it seems as though some of the class sizes somehow became negative, as though a
negative sign was incorrectly typed in front of them. Let's do a proc
freq for class size to see if this seems plausible.
proc freq data="c:\sasreg\elemapi";
tables acs_k3;
run;
avg class size k-3
Cumulative Cumulative
acs_k3 Frequency Percent Frequency Percent
-----------------------------------------------------------
-21 3 0.75 3 0.75
-20 2 0.50 5 1.26
-19 1 0.25 6 1.51
14 2 0.50 8 2.01
15 1 0.25 9 2.26
16 14 3.52 23 5.78
17 20 5.03 43 10.80
18 64 16.08 107 26.88
19 143 35.93 250 62.81
20 97 24.37 347 87.19
21 40 10.05 387 97.24
22 7 1.76 394 98.99
23 3 0.75 397 99.75
25 1 0.25 398 100.00
Frequency Missing = 2
Indeed, it seems that some of the class sizes somehow got negative signs put in front
of them. Let's look at the school and district number for these observations to see
if they come from the same district. Indeed, they all come from district 140.
proc print data="c:\sasreg\elemapi"; where (acs_k3 < 0); var snum dnum acs_k3; run;
Obs snum dnum acs_k3 38 600 140 -20 39 596 140 -19 40 611 140 -20 41 595 140 -21 42 592 140 -21 43 602 140 -21 85 116 294 . 306 4534 630 .Notice that when we looked at the observations where (acs_k3 < 0) this also included observations where acs_k3 is missing (represented as a period). To be more precise, the above command should exclude such observations like this.
Now, let's look at all of the observations for district 140.proc print data="c:\sasreg\elemapi"; where (acs_k3 < 0) and (acs_k3 ^= .); var snum dnum acs_k3; run;Obs snum dnum acs_k3 38 600 140 -20 39 596 140 -19 40 611 140 -20 41 595 140 -21 42 592 140 -21 43 602 140 -21
proc print data="c:\sasreg\elemapi"; where (dnum = 140); var snum dnum acs_k3; run;
Obs snum dnum acs_k3 38 600 140 -20 39 596 140 -19 40 611 140 -20 41 595 140 -21 42 592 140 -21 43 602 140 -21All of the observations from district 140 seem to have this problem. When you find such a problem, you want to go back to the original source of the data to verify the values. We have to reveal that we fabricated this error for illustration purposes, and that the actual data had no such problem. Let's pretend that we checked with district 140 and there was a problem with the data there, a hyphen was accidentally put in front of the class sizes making them negative. We will make a note to fix this! Let's continue checking our data.
Let's take a look at some graphical methods for inspecting data. For each variable, it is useful to inspect them using a histogram, boxplot, and stem-and-leaf plot. These graphs can show you information about the shape of your variables better than simple numeric statistics can. We already know about the problem with acs_k3, but let's see how these graphical methods would have revealed the problem with this variable.
First, we show a histogram for acs_k3. This shows us the observations where the average class size is negative.
proc univariate data="c:\sasreg\elemapi"; var acs_k3 ; histogram / cfill=gray; run;Likewise, a boxplot and stem-and-leaf plot would have called these observations to our attention as well. In SAS you can use the plot option with proc univariate to request a boxplot and stem and leaf plot. Below we show just the combined boxplot and stem and leaf plot from this output. You can see the outlying negative observations way at the bottom of the boxplot.
proc univariate data="c:\sasreg\elemapi" plot; var acs_k3; run;
Histogram # Boxplot
25+* 1 0
.** 10 |
.**************************** 137 +-----+
.****************************************** 207 *--+--*
.******* 34 |
.* 3 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.* 3 *
-21+* 3 *
----+----+----+----+----+----+----+----+--
* may represent up to 5 counts
We recommend plotting all of these graphs for the variables you will be analyzing. We
will omit, due to space considerations, showing these graphs for all of the variables.
However, in examining the variables, the stem-and-leaf plot for full seemed rather
unusual. Up to now, we have not seen anything problematic with this variable, but
look at the stem and leaf plot for full below. It shows 104 observations where the
percent with a full credential that is much lower than all other
observations. This is over 25% of the schools
and seems very unusual.
proc univariate data="c:\sasreg\elemapi" plot; var full; run;
Histogram # Boxplot
102.5+*************************** 81 |
.****************** 54 +-----+
92.5+**************** 46 | |
.************ 36 *-----*
82.5+********** 30 | |
.****** 17 | |
72.5+*** 8 | |
.** 6 | + |
62.5+** 4 | |
.** 5 | |
52.5+* 1 | |
.** 4 | |
42.5+* 3 | |
.* 1 | |
32.5+ | |
. | |
22.5+ | |
. | |
12.5+ | |
. | |
2.5+*********************************** 104 +-----+
----+----+----+----+----+----+----+
* may represent up to 3 counts
Let's look at the frequency distribution of full to see if we can understand
this better. The values go from 0.42 to 1.0, then jump to 37 and go up from there.
It appears as though some of the percentages are actually entered as proportions,
e.g., 0.42 was entered instead of 42 or 0.96 which really should have been 96.
proc freq data="c:\sasreg\elemapi" ;
tables full;
run;
pct full credential
Cumulative Cumulative
full Frequency Percent Frequency Percent
-----------------------------------------------------------------
0.4199999869 1 0.25 1 0.25
0.4499999881 1 0.25 2 0.50
0.4600000083 1 0.25 3 0.75
0.4699999988 1 0.25 4 1.00
0.4799999893 1 0.25 5 1.25
0.5 3 0.75 8 2.00
0.5099999905 1 0.25 9 2.25
0.5199999809 1 0.25 10 2.50
0.5299999714 1 0.25 11 2.75
0.5400000215 1 0.25 12 3.00
0.5600000024 2 0.50 14 3.50
0.5699999928 2 0.50 16 4.00
0.5799999833 1 0.25 17 4.25
0.5899999738 3 0.75 20 5.00
0.6000000238 1 0.25 21 5.25
0.6100000143 4 1.00 25 6.25
0.6200000048 2 0.50 27 6.75
0.6299999952 1 0.25 28 7.00
0.6399999857 3 0.75 31 7.75
0.6499999762 3 0.75 34 8.50
0.6600000262 2 0.50 36 9.00
0.6700000167 6 1.50 42 10.50
0.6800000072 2 0.50 44 11.00
0.6899999976 3 0.75 47 11.75
0.6999999881 1 0.25 48 12.00
0.7099999785 1 0.25 49 12.25
0.7200000286 2 0.50 51 12.75
0.7300000191 6 1.50 57 14.25
0.75 4 1.00 61 15.25
0.7599999905 2 0.50 63 15.75
0.7699999809 2 0.50 65 16.25
0.7900000215 3 0.75 68 17.00
0.8000000119 5 1.25 73 18.25
0.8100000024 8 2.00 81 20.25
0.8199999928 2 0.50 83 20.75
0.8299999833 2 0.50 85 21.25
0.8399999738 2 0.50 87 21.75
0.8500000238 3 0.75 90 22.50
0.8600000143 2 0.50 92 23.00
0.8999999762 3 0.75 95 23.75
0.9200000167 1 0.25 96 24.00
0.9300000072 1 0.25 97 24.25
0.9399999976 2 0.50 99 24.75
0.9499999881 2 0.50 101 25.25
0.9599999785 1 0.25 102 25.50
1 2 0.50 104 26.00
37 1 0.25 105 26.25
41 1 0.25 106 26.50
44 2 0.50 108 27.00
45 2 0.50 110 27.50
46 1 0.25 111 27.75
48 1 0.25 112 28.00
53 1 0.25 113 28.25
57 1 0.25 114 28.50
58 3 0.75 117 29.25
59 1 0.25 118 29.50
61 1 0.25 119 29.75
63 2 0.50 121 30.25
64 1 0.25 122 30.50
65 1 0.25 123 30.75
68 2 0.50 125 31.25
69 3 0.75 128 32.00
70 1 0.25 129 32.25
71 3 0.75 132 33.00
72 1 0.25 133 33.25
73 2 0.50 135 33.75
74 1 0.25 136 34.00
75 4 1.00 140 35.00
76 4 1.00 144 36.00
77 2 0.50 146 36.50
78 4 1.00 150 37.50
79 3 0.75 153 38.25
80 10 2.50 163 40.75
81 4 1.00 167 41.75
82 3 0.75 170 42.50
83 9 2.25 179 44.75
84 4 1.00 183 45.75
85 8 2.00 191 47.75
86 5 1.25 196 49.00
87 12 3.00 208 52.00
88 6 1.50 214 53.50
89 5 1.25 219 54.75
90 9 2.25 228 57.00
91 8 2.00 236 59.00
92 7 1.75 243 60.75
93 12 3.00 255 63.75
94 10 2.50 265 66.25
95 17 4.25 282 70.50
96 17 4.25 299 74.75
97 11 2.75 310 77.50
98 9 2.25 319 79.75
100 81 20.25 400 100.00
Let's see which district(s) these data came from.
proc freq data="c:\sasreg\elemapi" ;
where (full <= 1);
tables dnum;
run;
district number
Cumulative Cumulative
dnum Frequency Percent Frequency Percent
---------------------------------------------------------
401 104 100.00 104 100.00
We note that all 104 observations in which full was less than or equal to one
came from district 401. Let's see if this accounts for all of the
observations that come from district 401.
proc freq data="c:\sasreg\elemapi" ;
where (dnum = 401);
tables dnum;
run;
district number
Cumulative Cumulative
dnum Frequency Percent Frequency Percent
---------------------------------------------------------
401 104 100.00 104 100.00
All of the observations from this district seem to be recorded as proportions instead
of percentages. Again, let us state that this is a pretend problem that we inserted
into the data for illustration purposes. If this were a real life problem, we would
check with the source of the data and verify the problem. We will make a note to fix
this problem in the data as well.Another useful graphical technique for screening your data is a scatterplot matrix. While this is probably more relevant as a diagnostic tool searching for non-linearities and outliers in your data, it can also be a useful data screening tool, possibly revealing information in the joint distributions of your variables that would not be apparent from examining univariate distributions. Let's look at the scatterplot matrix for the variables in our regression model. This reveals the problems we have already identified, i.e., the negative class sizes and the percent full credential being entered as proportions.
proc insight data="c:\sasreg\elemapi"; scatter api00 acs_k3 meals full * api00 acs_k3 meals full; run;
We have identified three problems in our data. There are numerous missing values for meals, there were negatives accidentally inserted before some of the class sizes (acs_k3) and over a quarter of the values for full were proportions instead of percentages. The corrected version of the data is called elemapi2. Let's use that data file and repeat our analysis and see if the results are the same as our original analysis. First, let's repeat our original regression analysis below.
proc reg data="c:\sasreg\elemapi"
model api00 = acs_k3 meals full;
run;
Dependent Variable: api00 api 2000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 2634884 878295 213.41 <.0001
Error 309 1271713 4115.57673
Corrected Total 312 3906597
Root MSE 64.15276 R-Square 0.6745
Dependent Mean 596.40575 Adj R-Sq 0.6713
Coeff Var 10.75656
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 906.73916 28.26505 32.08 <.0001
acs_k3 avg class size k-3 1 -2.68151 1.39399 -1.92 0.0553
meals pct free meals 1 -3.70242 0.15403 -24.04 <.0001
full pct full credential 1 0.10861 0.09072 1.20 0.2321
Now, let's use the corrected data file and repeat the regression analysis. We see
quite a difference in the results! In the original analysis (above), acs_k3
was nearly significant, but in the corrected analysis (below) the results show this
variable to be not significant, perhaps due to the cases where class size was given a
negative value. Likewise, the percentage of teachers with full credentials was not
significant in the original analysis, but is significant in the corrected analysis,
perhaps due to the cases where the value was given as the proportion with full credentials
instead of the percent. Also, note that the corrected analysis is based on 398
observations instead of 313 observations, due to getting the complete data for the meals
variable which had lots of missing values.
proc reg data="c:\sasreg\elemapi2";
model api00 = acs_k3 meals full ;
run;
Dependent Variable: api00 api 2000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 6604966 2201655 615.55 <.0001
Error 394 1409241 3576.75370
Corrected Total 397 8014207
Root MSE 59.80597 R-Square 0.8242
Dependent Mean 648.46985 Adj R-Sq 0.8228
Coeff Var 9.22263
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 771.65811 48.86071 15.79 <.0001
acs_k3 avg class size k-3 1 -0.71706 2.23882 -0.32 0.7489
meals pct free meals 1 -3.68626 0.11178 -32.98 <.0001
full pct full credential 1 1.32714 0.23887 5.56 <.0001
From this point forward, we will use the corrected, elemapi2, data file.
So far we have covered some topics in data checking/verification, but we have not really discussed regression analysis itself. Let's now talk more about performing regression analysis in SAS.
1.3 Simple Linear Regression
Let's begin by showing some examples of simple linear regression using SAS. In this type of regression, we have only one predictor variable. This variable may be continuous, meaning that it may assume all values within a range, for example, age or height, or it may be dichotomous, meaning that the variable may assume only one of two values, for example, 0 or 1. The use of categorical variables with more than two levels will be covered in Chapter 3. There is only one response or dependent variable, and it is continuous.
In SAS, the dependent variable is listed immediately after the model statement followed by an equal sign and then one or more predictor variables. Let's examine the relationship between the size of school and academic performance to see if the size of the school is related to academic performance. For this example, api00 is the dependent variable and enroll is the predictor.
proc reg data="c:\sasreg\elemapi2"; model api00 = enroll; run;
Dependent Variable: api00 api 2000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 817326 817326 44.83 <.0001
Error 398 7256346 18232
Corrected Total 399 8073672
Root MSE 135.02601 R-Square 0.1012
Dependent Mean 647.62250 Adj R-Sq 0.0990
Coeff Var 20.84949
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 744.25141 15.93308 46.71 <.0001
enroll number of students 1 -0.19987 0.02985 -6.70 <.0001
Let's review this output a bit more carefully. First, we see that the F-test is
statistically significant, which means that the model is statistically significant. The
R-squared is .1012 means that approximately 10% of the variance of api00 is
accounted for by the model, in this case, enroll. The t-test for enroll
equals -6.70 , and is statistically significant, meaning that the regression coefficient
for enroll is significantly different from zero. Note that (-6.70)2 =
44.89, which is the same as the F-statistic (with some rounding error). The coefficient
for enroll is -.19987, or approximately -0.2, meaning that for a one unit increase
in enroll, we would expect a 0.2-unit decrease in api00. In other words,
a school with 1100 students would be expected to have an api score 20 units lower than a
school with 1000 students. The constant is 744.2514, and this is the
predicted value when enroll equals zero. In most cases, the
constant is not very interesting. We have prepared an annotated
output which shows the output from this regression along with an explanation of
each of the items in it.In addition to getting the regression table, it can be useful to see a scatterplot of the predicted and outcome variables with the regression line plotted. SAS makes this very easy for you by using the plot statement as part of proc reg. For example, below we show how to make a scatterplot of the outcome variable, api00 and the predictor, enroll. Note that the graph also includes the predicted values in the form of the regression line.
proc reg data="c:\sasreg\elemapi2"; model api00 = enroll ; plot api00 * enroll ; run;As you see, this one command produces a scatterplot and regression line, and it also includes the regression model with the correlation of the two variables in the title. We could include a 95% prediction interval using the pred option on the plot statement as illustrated below.
Another kind of graph that you might want to make is a residual versus fitted plot. As shown below, we can use the plot statement to make this graph. The keywords residual. and predicted. in this context refer to the residual value and predicted value from the regression analysis and can be abbreviated as r. and p. .proc reg data="c:\sasreg\elemapi2" ; model api00 = enroll ; plot api00 * enroll / pred; run; quit;
proc reg data="c:\sasreg\elemapi2"; model api00 = enroll ; plot residual. * predicted. ; run;
The table below shows a number of other keywords that can be used with the plot statement and the statistics they display.
| Keyword | Statistic |
| COOKD. | Cook's D influence statistics |
| COVRATIO. | standard influence of observation on covariance of betas |
| DFFITS. | standard influence of observation on predicted value |
| H. | leverage |
| LCL. | lower bound of
 %
confidence interval for individual prediction |
| LCLM. | lower bound of
%
confidence interval for the mean of the dependent variable |
| PREDICTED. | PRED. | P. | predicted values |
| PRESS. | residuals from refitting the model with current observation deleted |
| RESIDUAL. | R. | residuals |
| RSTUDENT. | studentized residuals with the current observation deleted |
| STDI. | standard error of the individual predicted value |
| STDP. | standard error of the mean predicted value |
| STDR. | standard error of the residual |
| STUDENT. | residuals divided by their standard errors |
| UCL. | upper bound of
%
confidence interval for individual prediction |
| UCLM. | upper bound of
%
confidence interval for the mean of the dependent variables |
Now, let's look at an example of multiple regression, in which we have one outcome (dependent) variable and multiple predictors. For this multiple regression example, we will regress the dependent variable, api00, on all of the predictor variables in the data set.
proc reg data="c:\sasreg\elemapi2" ;
model api00 = ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll ;
run;
Dependent Variable: api00 api 2000
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 9 6740702 748967 232.41 <.0001
Error 385 1240708 3222.61761
Corrected Total 394 7981410
Root MSE 56.76810 R-Square 0.8446
Dependent Mean 648.65063 Adj R-Sq 0.8409
Coeff Var 8.75172
Parameter Estimates
Parameter Standard
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 758.94179 62.28601 12.18 <.0001
ell english language learners 1 -0.86007 0.21063 -4.08 <.0001
meals pct free meals 1 -2.94822 0.17035 -17.31 <.0001
yr_rnd year round school 1 -19.88875 9.25844 -2.15 0.0323
mobility pct 1st year in school 1 -1.30135 0.43621 -2.98 0.0030
acs_k3 avg class size k-3 1 1.31870 2.25268 0.59 0.5586
acs_46 avg class size 4-6 1 2.03246 0.79832 2.55 0.0113
full pct full credential 1 0.60972 0.47582 1.28 0.2008
emer pct emer credential 1 -0.70662 0.60541 -1.17 0.2439
enroll number of students 1 -0.01216 0.01679 -0.72 0.4693
Let's examine the output from this regression analysis. As with the simple
regression, we look to the p-value of the F-test to see if the overall model is
significant. With a p-value of zero to four decimal places, the model is statistically
significant. The R-squared is 0.8446, meaning that approximately 84% of the variability of
api00 is accounted for by the variables in the model. In this case, the adjusted
R-squared indicates that about 84% of the variability of api00 is accounted for by
the model, even after taking into account the number of predictor variables in the model.
The coefficients for each of the variables indicates the amount of change one could expect
in api00 given a one-unit change in the value of that variable, given that all
other variables in the model are held constant. For example, consider the variable ell.
We would expect a decrease of 0.86 in the api00 score for every one unit
increase in ell, assuming that all other variables in the model are held
constant. The interpretation of much of the output from the multiple regression is
the same as it was for the simple regression. We have prepared an annotated output that more thoroughly explains the output
of this multiple regression analysis. You may be wondering what a 0.86 change in ell really means, and how you might compare the strength of that coefficient to the coefficient for another variable, say meals. To address this problem, we can use the stb option on the model statement to request that in addition to the standard output that SAS also display a table of the standardized values, sometimes called beta coefficients. Below we show just the portion of the output that includes these standardized values. The beta coefficients are used by some researchers to compare the relative strength of the various predictors within the model. Because the beta coefficients are all measured in standard deviations, instead of the units of the variables, they can be compared to one another. In other words, the beta coefficients are the coefficients that you would obtain if the outcome and predictor variables were all transformed to standard scores, also called z-scores, before running the regression.
proc reg data="c:\sasreg\elemapi2" ; model api00 = ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll / stb; run;
Parameter Estimates
Standardized
Variable Label DF Estimate
Intercept Intercept 1 0
ell english language learners 1 -0.14958
meals pct free meals 1 -0.66070
yr_rnd year round school 1 -0.05914
mobility pct 1st year in school 1 -0.06864
acs_k3 avg class size k-3 1 0.01273
acs_46 avg class size 4-6 1 0.05498
full pct full credential 1 0.06380
emer pct emer credential 1 -0.05801
enroll number of students 1 -0.01936
Because these standardized coefficients are all in the same standardized units you
can compare these coefficients to assess the relative strength of each of the
predictors. In this example, meals has the largest Beta coefficient, -0.66,
and acs_k3 has the smallest Beta, 0.013. Thus, a one standard deviation
increase in meals leads to a 0.66 standard deviation decrease in predicted api00,
with the other variables held constant. And, a one standard deviation increase in acs_k3,
in turn, leads to a 0.013 standard deviation increase api00 with the other
variables in the model held constant.In interpreting this output, remember that the difference between the regular coefficients (from the prior output) and the standardized coefficients above is the units of measurement. For example, to describe the raw coefficient for ell you would say "A one-unit decrease in ell would yield a .86-unit increase in the predicted api00." However, for the standardized coefficient (Beta) you would say, "A one standard deviation decrease in ell would yield a .15 standard deviation increase in the predicted api00."
So far, we have concerned ourselves with testing a single variable at a time, for example looking at the coefficient for ell and determining if that is significant. We can also test sets of variables, using the test command, to see if the set of variables are significant. First, let's start by testing a single variable, ell, using the test statement. Note that the part before the test command, test1:, is merely a label to identify the output of the test command. This label could be any short label to identify the output.
proc reg data="c:\sasreg\elemapi2" ; model api00 = ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll ; test1: test ell =0; run;
Test TEST1 Results for Dependent Variable api00
Mean
Source DF Square F Value Pr > F
Numerator 1 53732 16.67 <.0001
Denominator 385 3222.61761
If you compare this output with the output from the last regression you can see that
the result of the F-test, 16.67, is the same as the square of the result of the t-test in
the regression (-4.083^2 = 16.67). Note that you could get the same results if you typed
the following since SAS defaults to comparing the term(s) listed to 0.
proc reg data="c:\sasreg\elemapi2" ;
model api00 = ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll / stb;
test2: test ell;
run;
Test TEST2 Results for Dependent Variable api00
Mean
Source DF Square F Value Pr > F
Numerator 1 53732 16.67 <.0001
Denominator 385 3222.61761
Perhaps a more interesting test would be to see if the contribution of class size is
significant. Since the information regarding class size is contained in two
variables, acs_k3 and acs_46, so we include both of these
separated by a comma on the test
command. Below we show just the output from the test command.
proc reg data="c:\sasreg\elemapi2" ; model api00 = ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll ; test_class_size: test acs_k3, acs_46; run;
Test TEST_CLASS_SIZE Results for Dependent Variable api00
Mean
Source DF Square F Value Pr > F
Numerator 2 12742 3.95 0.0200
Denominator 385 3222.61761
The significant F-test, 3.95, means that the collective contribution of these two
variables is significant. One way to think of this, is that there is a significant
difference between a model with acs_k3 and acs_46 as compared to a model
without them, i.e., there is a significant difference between the "full" model
and the "reduced" models.Finally, as part of doing a multiple regression analysis you might be interested in seeing the correlations among the variables in the regression model. You can do this with proc corr as shown below.
proc corr data="c:\sasreg\elemapi2" ; var api00 ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll ; run;
Pearson Correlation Coefficients
Prob > |r| under H0: Rho=0
Number of Observations
api00 ell meals yr_rnd mobility
api00 1.00000 -0.76763 -0.90070 -0.47544 -0.20641
api 2000 <.0001 <.0001 <.0001 <.0001
400 400 400 400 399
ell -0.76763 1.00000 0.77238 0.49793 -0.02046
english language learners <.0001 <.0001 <.0001 0.6837
400 400 400 400 399
meals -0.90070 0.77238 1.00000 0.41845 0.21665
pct free meals <.0001 <.0001 <.0001 <.0001
400 400 400 400 399
yr_rnd -0.47544 0.49793 0.41845 1.00000 0.03479
year round school <.0001 <.0001 <.0001 0.4883
400 400 400 400 399
mobility -0.20641 -0.02046 0.21665 0.03479 1.00000
pct 1st year in school <.0001 0.6837 <.0001 0.4883
399 399 399 399 399
acs_k3 0.17100 -0.05565 -0.18797 0.02270 0.04014
avg class size k-3 0.0006 0.2680 0.0002 0.6517 0.4245
398 398 398 398 398
acs_46 0.23291 -0.17330 -0.21309 -0.04207 0.12769
avg class size 4-6 <.0001 0.0005 <.0001 0.4032 0.0110
397 397 397 397 396
full 0.57441 -0.48476 -0.52756 -0.39771 0.02521
pct full credential <.0001 <.0001 <.0001 <.0001 0.6156
400 400 400 400 399
emer -0.58273 0.47218 0.53304 0.43472 0.05961
pct emer credential <.0001 <.0001 <.0001 <.0001 0.2348
400 400 400 400 399
enroll -0.31817 0.40302 0.24103 0.59182 0.10502
number of students <.0001 <.0001 <.0001 <.0001 0.0360
400 400 400 400 399
Pearson Correlation Coefficients
Prob > |r| under H0: Rho=0
Number of Observations
acs_k3 acs_46 full emer enroll
api00 0.17100 0.23291 0.57441 -0.58273 -0.31817
api 2000 0.0006 <.0001 <.0001 <.0001 <.0001
398 397 400 400 400
ell -0.05565 -0.17330 -0.48476 0.47218 0.40302
english language learners 0.2680 0.0005 <.0001 <.0001 <.0001
398 397 400 400 400
meals -0.18797 -0.21309 -0.52756 0.53304 0.24103
pct free meals 0.0002 <.0001 <.0001 <.0001 <.0001
398 397 400 400 400
yr_rnd 0.02270 -0.04207 -0.39771 0.43472 0.59182
year round school 0.6517 0.4032 <.0001 <.0001 <.0001
398 397 400 400 400
mobility 0.04014 0.12769 0.02521 0.05961 0.10502
pct 1st year in school 0.4245 0.0110 0.6156 0.2348 0.0360
398 396 399 399 399
acs_k3 1.00000 0.27078 0.16057 -0.11033 0.10890
avg class size k-3 <.0001 0.0013 0.0277 0.0298
398 395 398 398 398
acs_46 0.27078 1.00000 0.11773 -0.12446 0.02829
avg class size 4-6 <.0001 0.0190 0.0131 0.5741
395 397 397 397 397
full 0.16057 0.11773 1.00000 -0.90568 -0.33769
pct full credential 0.0013 0.0190 <.0001 <.0001
398 397 400 400 400
emer -0.11033 -0.12446 -0.90568 1.00000 0.34309
pct emer credential 0.0277 0.0131 <.0001 <.0001
398 397 400 400 400
enroll 0.10890 0.02829 -0.33769 0.34309 1.00000
number of students 0.0298 0.5741 <.0001 <.0001
398 397 400 400 400
We can see that the strongest correlation with api00 is meals
with a correlation in excess of -0.9. The variables ell and emer
are also strongly correlated with api00.
All three of these correlations are negative, meaning that as the value of one variable
goes down, the value of the other variable tends to go up. Knowing that these variables
are strongly associated with api00, we might predict that they would be
statistically significant predictor variables in the regression model. Note that
the number of cases used for each correlation is determined on a
"pairwise" basis, for example there are 398 valid pairs of data for enroll
and acs_k3, so that correlation of .1089 is based on 398 observations.1.5 Transforming Variables
Earlier we focused on screening your data for potential errors. In the next chapter, we will focus on regression diagnostics to verify whether your data meet the assumptions of linear regression. Here, we will focus on the issue of normality. Some researchers believe that linear regression requires that the outcome (dependent) and predictor variables be normally distributed. We need to clarify this issue. In actuality, it is the residuals that need to be normally distributed. In fact, the residuals need to be normal only for the t-tests to be valid. The estimation of the regression coefficients do not require normally distributed residuals. As we are interested in having valid t-tests, we will investigate issues concerning normality.
A common cause of non-normally distributed residuals is non-normally distributed outcome and/or predictor variables. So, let us explore the distribution of our variables and how we might transform them to a more normal shape. Let's start by making a histogram of the variable enroll, which we looked at earlier in the simple regression.
proc univariate data="c:\sasreg\elemapi2"; var enroll ; histogram / cfill=gray; run;
We can use the normal option to superimpose a normal curve on this graph and the midpoints option to indicate that we want bins with midpoints from 100 to 1500 going in increments of 100.
Histograms are sensitive to the number of bins or columns that are used in the display. An alternative to histograms is the kernel density plot, which approximates the probability density of the variable. Kernel density plots have the advantage of being smooth and of being independent of the choice of origin, unlike histograms. You can add a kernel density plot to the above plot with he kernel option as illustrated below.proc univariate data="c:\sasreg\elemapi2"; var enroll ; histogram / cfill=gray normal midpoints=100 to 1500 by 100; run;
proc univariate data="c:\sasreg\elemapi2"; var enroll ; histogram / cfill=gray normal midpoints=100 to 1500 by 100 kernel; run;Not surprisingly, the kdensity plot also indicates that the variable enroll does not look normal.
There are two other types of graphs that are often used to examine the distribution of variables; quantile-quantile plots and normal probability plots.
A quantile-quantile plot graphs the quantiles of a variable against the quantiles of a normal (Gaussian) distribution. Such plots are sensitive to non-normality near the tails, and indeed we see considerable deviations from normal, the diagonal line, in the tails. This plot is typical of variables that are strongly skewed to the right.
proc univariate data="c:\sasreg\elemapi2"; var enroll ; qqplot / normal; run;The normal probability plot is also useful for examining the distribution of variables and is sensitive to deviations from normality nearer to the center of the distribution. We will use SAS proc capability to get the normal probability plot. Again, we see indications non-normality in enroll.
proc capability data="c:\sasreg\elemapi2" noprint; ppplot enroll ; run;
Given the skewness to the right in enroll, let us try a log transformation to see if that makes it more normal. Below we create a variable lenroll that is the natural log of enroll and then we repeat some of the above commands to see if lenroll is more normally distributed.
Now let's try showing a histogram for lenroll with a normal overlay and a kernel density estimate.data elemapi3; set "c:\sasreg\elemapi2"; lenroll = log(enroll); run;
We can see that lenroll looks quite normal. We could then create a quantile-quantile plot and a normal probability plot to further assess whether lenroll seems normal, as well as seeing how lenroll impacts the residuals, which is really the important consideration.proc univariate data=elemapi3 noprint; var lenroll ; histogram / cfill=grayd0 normal kernel (color = red); run;
1.6 Summary
In this lecture we have discussed the basics of how to perform simple and multiple regressions, the basics of interpreting output, as well as some related commands. We examined some tools and techniques for screening for bad data and the consequences such data can have on your results. Finally, we touched on the assumptions of linear regression and illustrated how you can check the normality of your variables and how you can transform your variables to achieve normality. The next chapter will pick up where this chapter has left off, going into a more thorough discussion of the assumptions of linear regression and how you can use SAS to assess these assumptions for your data. In particular, the next lecture will address the following issues.
- Checking for points that exert undue influence on the coefficients
- Checking for constant error variance (homoscedasticity)
- Checking for linear relationships
- Checking model specification
- Checking for multicollinearity
- Checking normality of residuals
- Related Web Pages
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