Regression with SPSS Chapter 1 – Simple and Multiple Regression

We see that we have 400 observations for most of our variables, but some variables have missing values, like meals which has a valid N of 315.  Note that when we did our original regression analysis the DF TOTAL was 312, implying only 313 of the observations were included in the analysis.  But, the descriptives command suggests we have 400 observations in our data file. 

Let's examine the output more carefully for the variables we used in our regression analysis above, namely api00, acs_k3, meals, full, and yr_rnd. For api00, we see that the values range from 369 to 940 and there are 400 valid values.  For acs_k3, the average class size ranges from -21 to 25 and there are 2 missing values.  An average class size of -21 sounds wrong, and later we will investigate this further.  The variable meals ranges from 6% getting free meals to 100% getting free meals, so these values seem reasonable, but there are only 315 valid values for this variable.  The percent of teachers being full credentialed ranges from .42 to 100, and all of the values are valid.  The variable yr_rnd ranges from 0 to 1 (which makes sense since this is a dummy variable) and all values are valid.  

examine
  /variables=acs_k3
  /plot histogram stem boxplot . 

Case Processing Summary

	Cases
	Valid		Missing		Total
	N	Percent	N	Percent	N	Percent
ACS_K3	398	99.5%	2	.5%	400	100.0%




Descriptives

			Statistic	Std. Error
ACS_K3	Mean		18.55	.251
	95% Confidence Interval for Mean	Lower Bound	18.05
		Upper Bound	19.04
	5% Trimmed Mean		19.13
	Median		19.00
	Variance		25.049
	Std. Deviation		5.005
	Minimum		-21
	Maximum		25
	Range		46
	Interquartile Range		2.00
	Skewness		-7.106	.122
	Kurtosis		53.014	.244

avg class size k-3 Stem-and-Leaf Plot

 Frequency    Stem &  Leaf

     9.00 Extremes    (=<15.0)
    14.00       16 .  00000
      .00       16 .
    20.00       17 .  0000000
      .00       17 .
    64.00       18 .  000000000000000000000
      .00       18 .
   143.00       19 .  000000000000000000000000000000000000000000000000
      .00       19 .
    97.00       20 .  00000000000000000000000000000000
      .00       20 .
    40.00       21 .  0000000000000
      .00       21 .
     7.00       22 .  00
     4.00 Extremes    (>=23.0)

 Stem width:     1
 Each leaf:       3 case(s)

 We see that the histogram and boxplot are effective in showing the schools with class sizes that are negative.  The stem and leaf plot indicates that there are some "Extremes" that are less than 16, but it does not reveal how extreme these values are.  Looking at the boxplot and histogram we see observations where the class sizes are around -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 frequencies for class size to see if this seems plausible.

frequencies
  /var acs_k3.

Statistics
ACS_K3

N	Valid	398
	Missing	2




ACS_K3

		Frequency	Percent	Valid Percent	Cumulative Percent
Valid	-21	3	.8	.8	.8
	-20	2	.5	.5	1.3
	-19	1	.3	.3	1.5
	14	2	.5	.5	2.0
	15	1	.3	.3	2.3
	16	14	3.5	3.5	5.8
	17	20	5.0	5.0	10.8
	18	64	16.0	16.1	26.9
	19	143	35.8	35.9	62.8
	20	97	24.3	24.4	87.2
	21	40	10.0	10.1	97.2
	22	7	1.8	1.8	99.0
	23	3	.8	.8	99.7
	25	1	.3	.3	100.0
	Total	398	99.5	100.0
Missing	System	2	.5
Total		400	100.0

 

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.  

compute filtvar = (acs_k3 < 0).
filter by filtvar.
list cases
  /var snum dnum acs_k3.
filter off.

     SNUM    DNUM ACS_K3
      600     140   -20
      596     140   -19
      611     140   -20
      595     140   -21
      592     140   -21
      602     140   -21

Now,  let's look at all of the observations for district 140.

compute filtvar = (dnum = 140).
filter by filtvar.
list cases
  /var snum dnum acs_k3.
filter off.

     SNUM    DNUM ACS_K3
      600     140   -20
      596     140   -19
      611     140   -20
      595     140   -21
      592     140   -21
      602     140   -21

Number of cases read:  6    Number of cases listed:  6

All 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.

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 histogram for full seemed rather unusual.  Up to now, we have not seen anything problematic with this variable, but look at the histogram for full below. It shows over 100 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.

frequencies
  variables=full
  /format=notable
  /histogram .
 

Statistics
FULL

N	Valid	400
	Missing	0

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.

frequencies
  variables=full  . 

Statistics
FULL

N	Valid	400
	Missing	0




FULL

		Frequency	Percent	Valid Percent	Cumulative Percent
Valid	.42	1	.3	.3	.3
	.45	1	.3	.3	.5
	.46	1	.3	.3	.8
	.47	1	.3	.3	1.0
	.48	1	.3	.3	1.3
	.50	3	.8	.8	2.0
	.51	1	.3	.3	2.3
	.52	1	.3	.3	2.5
	.53	1	.3	.3	2.8
	.54	1	.3	.3	3.0
	.56	2	.5	.5	3.5
	.57	2	.5	.5	4.0
	.58	1	.3	.3	4.3
	.59	3	.8	.8	5.0
	.60	1	.3	.3	5.3
	.61	4	1.0	1.0	6.3
	.62	2	.5	.5	6.8
	.63	1	.3	.3	7.0
	.64	3	.8	.8	7.8
	.65	3	.8	.8	8.5
	.66	2	.5	.5	9.0
	.67	6	1.5	1.5	10.5
	.68	2	.5	.5	11.0
	.69	3	.8	.8	11.8
	.70	1	.3	.3	12.0
	.71	1	.3	.3	12.3
	.72	2	.5	.5	12.8
	.73	6	1.5	1.5	14.3
	.75	4	1.0	1.0	15.3
	.76	2	.5	.5	15.8
	.77	2	.5	.5	16.3
	.79	3	.8	.8	17.0
	.80	5	1.3	1.3	18.3
	.81	8	2.0	2.0	20.3
	.82	2	.5	.5	20.8
	.83	2	.5	.5	21.3
	.84	2	.5	.5	21.8
	.85	3	.8	.8	22.5
	.86	2	.5	.5	23.0
	.90	3	.8	.8	23.8
	.92	1	.3	.3	24.0
	.93	1	.3	.3	24.3
	.94	2	.5	.5	24.8
	.95	2	.5	.5	25.3
	.96	1	.3	.3	25.5
	1.00	2	.5	.5	26.0
	37.00	1	.3	.3	26.3
	41.00	1	.3	.3	26.5
	44.00	2	.5	.5	27.0
	45.00	2	.5	.5	27.5
	46.00	1	.3	.3	27.8
	48.00	1	.3	.3	28.0
	53.00	1	.3	.3	28.3
	57.00	1	.3	.3	28.5
	58.00	3	.8	.8	29.3
	59.00	1	.3	.3	29.5
	61.00	1	.3	.3	29.8
	63.00	2	.5	.5	30.3
	64.00	1	.3	.3	30.5
	65.00	1	.3	.3	30.8
	68.00	2	.5	.5	31.3
	69.00	3	.8	.8	32.0
	70.00	1	.3	.3	32.3
	71.00	3	.8	.8	33.0
72.00	1	.3	.3	33.3
73.00	2	.5	.5	33.8
74.00	1	.3	.3	34.0
75.00	4	1.0	1.0	35.0
76.00	4	1.0	1.0	36.0
77.00	2	.5	.5	36.5
78.00	4	1.0	1.0	37.5
79.00	3	.8	.8	38.3
80.00	10	2.5	2.5	40.8
81.00	4	1.0	1.0	41.8
82.00	3	.8	.8	42.5
83.00	9	2.3	2.3	44.8
84.00	4	1.0	1.0	45.8
85.00	8	2.0	2.0	47.8
86.00	5	1.3	1.3	49.0
87.00	12	3.0	3.0	52.0
88.00	6	1.5	1.5	53.5
89.00	5	1.3	1.3	54.8
90.00	9	2.3	2.3	57.0
91.00	8	2.0	2.0	59.0
92.00	7	1.8	1.8	60.8
93.00	12	3.0	3.0	63.8
94.00	10	2.5	2.5	66.3
95.00	17	4.3	4.3	70.5
96.00	17	4.3	4.3	74.8
97.00	11	2.8	2.8	77.5
98.00	9	2.3	2.3	79.8
100.00	81	20.3	20.3	100.0
Total	400	100.0	100.0

 

Let's see which district(s) these data came from. 

compute filtvar = (full < 1).
filter by filtvar.
frequencies
  variables=dnum  . 
filter off.

Statistics
DNUM

N	Valid	102
	Missing	0




DNUM

		Frequency	Percent	Valid Percent	Cumulative Percent
Valid	401	102	100.0	100.0	100.0

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.

compute filtvar = (dnum = 401).
filter by filtvar.
frequencies
  variables=dnum  .
filter off. 

Statistics
DNUM

N	Valid	104
	Missing	0




DNUM

		Frequency	Percent	Valid Percent	Cumulative Percent
Valid	401	104	100.0	100.0	100.0

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 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, but 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. 

graph
  /scatterplot(matrix)=acs_46 acs_k3 api00 api99 .

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. But first, let's repeat our original regression analysis below.

regression
  /dependent api00
  /method=enter acs_k3 meals full.
 

<some output omitted to save space>

Coefficients(a)

		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
Model		B	Std. Error	Beta
1	(Constant)	906.739	28.265		32.080	.000
	ACS_K3	-2.682	1.394	-.064	-1.924	.055
	MEALS	-3.702	.154	-.808	-24.038	.000
	FULL	.109	.091	.041	1.197	.232
a Dependent Variable: API00

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 (which was revealed in the deleted output), due to getting the complete data for the meals variable which had lots of missing values. 

get file = "c:spssregelemapi2.sav".

regression
  /dependent api00
  /method=enter acs_k3 meals full.

<some output omitted to save space>





Coefficients(a)

		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
Model		B	Std. Error	Beta
1	(Constant)	771.658	48.861		15.793	.000
	ACS_K3	-.717	2.239	-.007	-.320	.749
	MEALS	-3.686	.112	-.828	-32.978	.000
	FULL	1.327	.239	.139	5.556	.000
a Dependent Variable: API00

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 SPSS.

1.3 Simple Linear Regression

Let's begin by showing some examples of simple linear regression using SPSS. 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.

When using SPSS for simple regression, the dependent variable is given in the /dependent subcommand and the predictor is given after the /method=enter subcommand. 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.

regression
  /dependent api00
  /method=enter enroll.
 

Variables Entered/Removed(b)

Model	Variables Entered	Variables Removed	Method
1	ENROLL(a)	.	Enter
a All requested variables entered.
b Dependent Variable: API00




Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.318(a)	.101	.099	135.026
a Predictors: (Constant), ENROLL




ANOVA(b)

Model		Sum of Squares	df	Mean Square	F	Sig.
1	Regression	817326.293	1	817326.293	44.829	.000(a)
	Residual	7256345.704	398	18232.024
	Total	8073671.997	399
a Predictors: (Constant), ENROLL
b Dependent Variable: API00







Coefficients(a)

		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
Model		B	Std. Error	Beta
1	(Constant)	744.251	15.933		46.711	.000
	ENROLL	-.200	.030	-.318	-6.695	.000
a Dependent Variable: API00

 

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 .101 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 -6.695 , and is statistically significant, meaning that the regression coefficient for enroll is significantly different from zero. Note that (-6.695)² = -44.82, which is the same as the F-statistic (with some rounding error). The coefficient for enroll is -.200, meaning that for a one unit increase in enroll, we would expect a .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.  You can do this with the graph command as shown below.  However, by default, SPSS does not include a regression line and the only way we know to include it is by clicking on the graph and from the pulldown menus choosing Chart then Options and then clicking on the checkbox fit line total to add the regression line.  The graph below is what you see after adding the regression line to the graph.

graph
  /scatterplot(bivar)=enroll with api00
  /missing=listwise .

Another kind of graph that you might want to make is a residual versus fitted plot.  As shown below, we can use the /scatterplot subcommand as part of the regress command to make this graph.  The keywords *zresid and *adjpred in this context refer to the residual value and predicted value from the regression analysis.

regression
  /dependent api00
  /method=enter enroll
  /scatterplot=(*zresid ,*adjpred ) .

<output deleted to save space> 

The table below shows a number of other keywords that can be used with the /scatterplot subcommand and the statistics they display.

1.4 Multiple Regression

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.

regression
  /dependent api00
  /method=enter ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll .
 

Variables Entered/Removed(b)

Model	Variables Entered	Variables Removed	Method
1	ENROLL, ACS_46, MOBILITY, ACS_K3,  EMER, ELL, YR_RND, MEALS, FULL(a)	.	Enter
a All requested variables entered.
b Dependent Variable: API00




Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of  the Estimate
1	.919(a)	.845	.841	56.768
a Predictors: (Constant), ENROLL, ACS_46, MOBILITY, ACS_K3, EMER,  ELL, YR_RND, MEALS, FULL




ANOVA(b)

Model		Sum of  Squares	df	Mean  Square	F	Sig.
1	Regression	6740702.006	9	748966.890	232.409	.000(a)
	Residual	1240707.781	385	3222.618
	Total	7981409.787	394
a Predictors: (Constant), ENROLL, ACS_46, MOBILITY,  ACS_K3, EMER, ELL, YR_RND, MEALS, FULL
b Dependent Variable: API00







Coefficients(a)

		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
Model		B	Std. Error	Beta
1	(Constant)	758.942	62.286		12.185	.000
	ELL	-.860	.211	-.150	-4.083	.000
	MEALS	-2.948	.170	-.661	-17.307	.000
	YR_RND	-19.889	9.258	-.059	-2.148	.032
	MOBILITY	-1.301	.436	-.069	-2.983	.003
	ACS_K3	1.319	2.253	.013	.585	.559
	ACS_46	2.032	.798	.055	2.546	.011
	FULL	.610	.476	.064	1.281	.201
	EMER	-.707	.605	-.058	-1.167	.244
	ENROLL	-1.216E-02	.017	-.019	-.724	.469
a Dependent Variable: API00

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 three decimal places, the model is statistically significant. The R-squared is 0.845, meaning that approximately 85% 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 refer to the column of Beta coefficients, also known as standardized regression coefficients.  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. In this example, meals has the largest Beta coefficient, -0.661, and acs_k3 has the smallest Beta, 0.013.  Thus, a one standard deviation increase in meals leads to a 0.661 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 and the standardized coefficients 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 test on the /method subcommand, to see if the set of variables is significant.  First, let's start by testing a single variable, ell, using the /method=test subcommand.  Note that we have two /method subcommands, the first including all of the variables we want, except for ell, using /method=enter .  Then, the second subcommand uses /method=test(ell) to indicate that we wish to test the effect of adding ell to the model previously specified.

As you see in the output below, SPSS forms two models, the first with all of the variables specified in the first /model subcommand that indicates that the 8 variables in the first model are significant (F=249.256).  Then, SPSS adds ell to the model and reports an F test evaluating the addition of the variable ell, with an F value of 16.673 and a p value of 0.000, indicating that the addition of ell is significant.  Then, SPSS reports the significance of the overall model with all 9 variables, and the F value for that is  232.4 and is significant.

regression
  /dependent api00
  /method=enter meals yr_rnd mobility acs_k3 acs_46 full emer enroll
  /method=test(ell).

Variables Entered/Removed(b)

Model	Variables Entered	Variables Removed	Method
1	ENROLL, ACS_46, MOBILITY, ACS_K3,  EMER, MEALS, YR_RND, FULL(a)	.	Enter
2	ELL	.	Test
a All requested variables entered.
b Dependent Variable: API00

Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.915(a)	.838	.834	57.909
2	.919(b)	.845	.841	56.768
a Predictors: (Constant), ENROLL, ACS_46, MOBILITY, ACS_K3,  EMER, MEALS, YR_RND, FULL
b Predictors: (Constant), ENROLL, ACS_46, MOBILITY, ACS_K3,  EMER, MEALS, YR_RND, FULL, ELL

ANOVA(d)

Model			Sum of  Squares	df	Mean  Square	F	Sig.	R Square Change
1	Regression		6686970.454		835871.307	249.256	.000(a)
	Residual		1294439.333	386	3353.470
	Total		7981409.787	394
2	Subset Tests	ELL	53731.552	1	53731.552	16.673	.000(b)	.007
	Regression		6740702.006	9	748966.890	232.409	.000(c)
	Residual		1240707.781	385	3222.618
	Total		7981409.787	394
a Predictors: (Constant), ENROLL, ACS_46, MOBILITY, ACS_K3,  EMER, MEALS, YR_RND, FULL
b Tested against the full model.
c Predictors in the Full Model: (Constant), ENROLL, ACS_46, MOBILITY,  ACS_K3, EMER, MEALS, YR_RND, FULL, ELL.
d Dependent Variable: API00

Coefficients(a)

		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
Model		B	Std. Error	Beta
1	(Constant)	779.331	63.333		12.305	.000
	MEALS	-3.447	.121	-.772	-28.427	.000
	YR_RND	-24.029	9.388	-.071	-2.560	.011
	MOBILITY	-.728	.421	-.038	-1.728	.085
	ACS_K3	.178	2.280	.002	.078	.938
	ACS_46	2.097	.814	.057	2.575	.010
	FULL	.632	.485	.066	1.301	.194
	EMER	-.670	.618	-.055	-1.085	.279
	ENROLL	-3.092E-02	.016	-.049	-1.876	.061
2	(Constant)	758.942	62.286		12.185	.000
	MEALS	-2.948	.170	-.661	-17.307	.000
	YR_RND	-19.889	9.258	-.059	-2.148	.032
	MOBILITY	-1.301	.436	-.069	-2.983	.003
	ACS_K3	1.319	2.253	.013	.585	.559
	ACS_46	2.032	.798	.055	2.546	.011
	FULL	.610	.476	.064	1.281	.201
	EMER	-.707	.605	-.058	-1.167	.244
	ENROLL	-1.216E-02	.017	-.019	-.724	.469
	ELL	-.860	.211	-.150	-4.083	.000
a Dependent Variable: API00

Excluded Variables(b)

		Beta In	t	Sig.	Partial Correlation	Collinearity  Statistics
Model						Tolerance
1	ELL	-.150(a)	-4.083	.000	-.204	.301
a Predictors in the Model: (Constant), ENROLL, ACS_46, MOBILITY,  ACS_K3, EMER, MEALS, YR_RND, FULL
b Dependent Variable: API00

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 in the parentheses of the method-test( ) command.  The output below shows the F value for this test is 3.954 with a p value of 0.020, indicating that the overall 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.

regression
  /dependent api00
  /method=enter ell meals yr_rnd mobility full emer enroll
  /method=test(acs_k3 acs_46).
 

Variables Entered/Removed(b)

Model	Variables Entered	Variables Removed	Method
1	ENROLL, MOBILITY, MEALS,  EMER, YR_RND, ELL, FULL(a)	.	Enter
2	ACS_46, ACS_K3	.	Test
a All requested variables entered.
b Dependent Variable: API00




Model Summary

Model	R	R Square	Adjusted R Square	Std. Error of the Estimate
1	.917(a)	.841	.838	57.200
2	.919(b)	.845	.841	56.768
a Predictors: (Constant), ENROLL, MOBILITY, MEALS,  EMER, YR_RND, ELL, FULL
b Predictors: (Constant), ENROLL, MOBILITY, MEALS,  EMER, YR_RND, ELL, FULL, ACS_46, ACS_K3




ANOVA(d)

Model			Sum of  Squares	df	Mean  Square	F	Sig.	R Square  Change
1	Regression		6715217.454	7	959316.779	293.206	.000(a)
	Residual		1266192.333	387	3271.815
	Total		7981409.787	394
2	Subset  Tests	ACS_K3,  ACS_46	25484.552	2	12742.276	3.954	.020(b)	.003
	Regression		6740702.006	9	748966.890	232.409	.000(c)
	Residual		1240707.781	385	3222.618
	Total		7981409.787	394
a Predictors: (Constant), ENROLL, MOBILITY, MEALS,  EMER, YR_RND, ELL, FULL
b Tested against the full model.
c Predictors in the Full Model: (Constant), ENROLL, MOBILITY, MEALS,  EMER, YR_RND, ELL, FULL, ACS_46, ACS_K3.
d Dependent Variable: API00




Coefficients(a)

		Unstandardized Coefficients		Standardized Coefficients	t	Sig.
Model		B	Std. Error	Beta
1	(Constant)	846.223	48.053		17.610	.000
	ELL	-.840	.211	-.146	-3.988	.000
	MEALS	-3.040	.167	-.681	-18.207	.000
	YR_RND	-18.818	9.321	-.056	-2.019	.044
	MOBILITY	-1.075	.432	-.057	-2.489	.013
	FULL	.589	.474	.062	1.242	.215
	EMER	-.763	.606	-.063	-1.258	.209
	ENROLL	-9.527E-03	.017	-.015	-.566	.572
2	(Constant)	758.942	62.286		12.185	.000
	ELL	-.860	.211	-.150	-4.083	.000
	MEALS	-2.948	.170	-.661	-17.307	.000
	YR_RND	-19.889	9.258	-.059	-2.148	.032
	MOBILITY	-1.301	.436	-.069	-2.983	.003
	FULL	.610	.476	.064	1.281	.201
	EMER	-.707	.605	-.058	-1.167	.244
	ENROLL	-1.216E-02	.017	-.019	-.724	.469
	ACS_K3	1.319	2.253	.013	.585	.559
	ACS_46	2.032	.798	.055	2.546	.011
a Dependent Variable: API00



Tolerance

Excluded Variables(b)

		Beta In	t	Sig.	Partial Correlation	Collinearity Statistics
Model
1	ACS_K3	.025(a)	1.186	.236	.060	.900
	ACS_46	.058(a)	2.753	.006	.139	.913
a Predictors in the Model: (Constant), ENROLL, MOBILITY, MEALS,  EMER, YR_RND, ELL, FULL
b Dependent Variable: API00

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 the correlations command as shown below.

correlations
  /variables=api00 ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll.

We can see that the strongest correlation with api00 is meals with a correlation in excess of -.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.  In this section 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.