multiple regression ii
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Multiple Regression II. Fenster. Multiple Regression. Let’s go through an example using multiple regression and compare results between simple regression and multiple regression. Teacher Salary Hypothesis. Let’s say one hypothesized that: - PowerPoint PPT PresentationTRANSCRIPT
Multiple Regression II
Fenster
Multiple Regression Let’s go through an example using multiple
regression and compare results between simple regression and multiple regression.
Teacher Salary Hypothesis Let’s say one hypothesized that: H1: The higher the teacher salary in a
county, the better students performed on state mandated assessments.
Teacher Salary Hypothesis The researcher was interested in studying
the relationship between teacher salary and student performance on state mandated assessments at the county level.
Unit of analysis is county. Since the researcher lives in FL, she
chose to collect data on that state.
Teacher Salary Hypothesis So there are 67 counties in FL. DSSMATH is a state mandated
assessment that can be used to measure yearly progress in math for the NCLB act.
DSSREA is a state mandated assessment that can be used to measure yearly progress in reading for the NCLB act.
Univariate AnalysisStatistics
67 67 67 67
18 18 18 18
1440.5970 1501.3284 36.1378 41.3897
1443.0000 1505.0000 35.8490 41.9300
57.05581 70.52453 2.92448 12.42106
-.344 -.302 .902 -.042
.293 .293 .293 .293
.378 -.128 1.258 .318
.578 .578 .578 .578
1404.0000 1458.0000 34.3820 33.7900
1443.0000 1505.0000 35.8490 41.9300
1473.0000 1548.0000 37.0620 50.9600
Valid
Missing
N
Mean
Median
Std. Deviation
Skewness
Std. Error of Skewness
Kurtosis
Std. Error of Kurtosis
25
50
75
Percentiles
DSSMATH DSSREA
Teachersalary in1000s ofdollars FRL
Univariate Analysis
DSSMATH
1540.0
1520.0
1500.0
1480.0
1460.0
1440.0
1420.0
1400.0
1380.0
1360.0
1340.0
1320.0
1300.0
1280.0
DSSMATH
Fre
qu
en
cy
12
10
8
6
4
2
0
Std. Dev = 57.06
Mean = 1440.6
N = 67.00
Univariate Analysis
DSSREA
DSSREA
Fre
qu
en
cy
12
10
8
6
4
2
0
Std. Dev = 70.52
Mean = 1501.3
N = 67.00
Univariate Analysis
Teacher salary in 1000s of dollars
45.0
44.0
43.0
42.0
41.0
40.0
39.0
38.0
37.0
36.0
35.0
34.0
33.0
32.0
31.0
30.0
Teacher salary in 1000s of dollars
Fre
qu
en
cy
14
12
10
8
6
4
2
0
Std. Dev = 2.92
Mean = 36.1
N = 67.00
Univariate Analysis
Percentage of Students on free or reduced lunch
70.0
65.0
60.0
55.0
50.0
45.0
40.0
35.0
30.0
25.0
20.0
15.0
10.0
Percentage of Students on free or reduced lunch
Fre
qu
en
cy
20
10
0
Std. Dev = 12.42
Mean = 41.4
N = 67.00
Univariate Analysis I would conclude that all of my variables are
at least “reasonably” normally distributed.
Pearson Product Moment Correlations on the data
Did we find support for H1?
Correlations
1 .929** .255*
. .000 .018
67 67 67
.929** 1 .154
.000 . .106
67 67 67
.255* .154 1
.018 .106 .
67 67 67
Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N
DSSMATH
DSSREA
Teacher salary in1000s of dollars
DSSMATH DSSREA
Teachersalary in1000s ofdollars
Correlation is significant at the 0.01 level (1-tailed).**.
Correlation is significant at the 0.05 level (1-tailed).*.
Spearman’s rho correlations on the data
Correlations
1.000 .911** .294**
. .000 .008
67 67 67
.911** 1.000 .166
.000 . .090
67 67 67
.294** .166 1.000
.008 .090 .
67 67 67
Correlation Coefficient
Sig. (1-tailed)
N
Correlation Coefficient
Sig. (1-tailed)
N
Correlation Coefficient
Sig. (1-tailed)
N
DSSMATH
DSSREA
Teacher salary in1000s of dollars
Spearman's rhoDSSMATH DSSREA
Teachersalary in1000s ofdollars
Correlation is significant at the .01 level (1-tailed).**.
Regression and Pearson correlations essentially the same test
We can get the same result in simple regression that we got with the Pearson Product Moment correlation (assuming we use the same set of data).
Results for simple regression: MathDescriptive Statistics
1440.5970 57.05581 67
36.1378 2.92448 67
DSSMATH
Teacher salary in1000s of dollars
Mean Std. Deviation N
Variables Entered/Removedb
Teachersalary in1000s ofdollars
a. Enter
Model1
VariablesEntered
VariablesRemoved Method
All requested variables entered.a.
Dependent Variable: DSSMATHb.
Results for simple regression: MathModel Summaryb
.255a .065 .051 55.58546Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), Teacher salary in 1000s ofdollars
a.
Dependent Variable: DSSMATHb.
ANOVAb
14020.806 1 14020.806 4.538 .037a
200833.3 65 3089.743
214854.1 66
Regression
Residual
Total
Model1
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), Teacher salary in 1000s of dollarsa.
Dependent Variable: DSSMATHb.
Results for simple regression: Math
The “sig” we see on the SPSS results page represents a two-tailed probability value. We should divide that probability value in ½ to give us a one-tailed probablity.
Coefficientsa
1260.491 84.820 14.861 .000
4.984 2.340 .255 2.130 .037
(Constant)
Teacher salary in1000s of dollars
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: DSSMATHa.
Results for simple regression: Math Can we reject the null hypothesis for H1? What probability value did we get for the
relationship between teacher salary and DSS MATH when using correlation? Answer .018
What probability value did we get for the relationship between teacher salary and DSS MATH when using correlation regression? Answer .037/2=.018
Results for simple regression: MathCasewise Diagnosticsa
-2.340 1312.00 1442.0882 -130.0882
-2.593 1284.00 1428.1234 -144.1234
-2.388 1296.00 1428.7215 -132.7215
Case Number22
33
40
Std. Residual DSSMATHPredicted
Value Residual
Dependent Variable: DSSMATHa.
Results for simple regression: MathResiduals Statisticsa
1411.7365 1484.5854 1440.5970 14.57520 67
-144.1234 99.9684 .0000 55.16275 67
-1.980 3.018 .000 1.000 67
-2.593 1.798 .000 .992 67
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum Maximum Mean Std. Deviation N
Dependent Variable: DSSMATHa.
Results for simple regression: Math
Teacher salary in 1000s of dollars
464442403836343230
DS
SM
AT
H
1600
1500
1400
1300
1200
Simple Regression Results for Reading
Descriptive Statistics
1501.3284 70.52453 67
36.1378 2.92448 67
DSSREA
Teacher salary in1000s of dollars
Mean Std. Deviation N
Variables Entered/Removedb
Teachersalary in1000s ofdollars
a. Enter
Model1
VariablesEntered
VariablesRemoved Method
All requested variables entered.a.
Dependent Variable: DSSREAb.
Simple Regression Results for Reading
Model Summaryb
.154a .024 .009 70.21537Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), Teacher salary in 1000s ofdollars
a.
Dependent Variable: DSSREAb.
ANOVAb
7801.938 1 7801.938 1.582 .213a
320462.8 65 4930.198
328264.8 66
Regression
Residual
Total
Model1
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), Teacher salary in 1000s of dollarsa.
Dependent Variable: DSSREAb.
Simple Regression Results for Reading
Coefficientsa
1366.977 107.144 12.758 .000
3.718 2.955 .154 1.258 .213
(Constant)
Teacher salary in1000s of dollars
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: DSSREAa.
Simple Regression Results for Reading
Casewise Diagnosticsa
-2.535 1314.00 1492.0236 -178.0236Case Number33
Std. Residual DSSREAPredicted
Value Residual
Dependent Variable: DSSREAa.
Residuals Statisticsa
1479.7996 1534.1420 1501.3284 10.87250 67
-178.0235 140.3475 .0000 69.68140 67
-1.980 3.018 .000 1.000 67
-2.535 1.999 .000 .992 67
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum Maximum Mean Std. Deviation N
Dependent Variable: DSSREAa.
Simple Regression Results for Reading Can we reject the null hypothesis for H1 when it
comes to reading? What probability value did we get for the
relationship between teacher salary and DSS REA when using correlation? Answer .106
What probability value did we get for the relationship between teacher salary and DSS REA when using correlation regression? Answer .213/2=.106WE FAIL TO REJECT THE NULL FOR READING!
Multiple Regression Results for ReadingDescriptive Statistics
1501.3284 70.52453 67
36.1378 2.92448 67
41.3897 12.42106 67
DSSREA
Teacher salary in 1000sof dollars
Percentage of Studentson free or reduced lunch
Mean Std. Deviation N
Variables Entered/Removedb
Percentage ofStudentson free orreducedlunch,Teachersalary in1000s ofdollars
a
. Enter
Model1
VariablesEntered
VariablesRemoved Method
All requested variables entered.a.
Dependent Variable: DSSREAb.
Multiple Regression Results for ReadingModel Summaryb
.620a .385 .366 56.16920Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), Percentage of Students on freeor reduced lunch, Teacher salary in 1000s of dollars
a.
Dependent Variable: DSSREAb. ANOVAb
126346.1 2 63173.067 20.023 .000a
201918.6 64 3154.979
328264.8 66
Regression
Residual
Total
Model1
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), Percentage of Students on free or reduced lunch, Teachersalary in 1000s of dollars
a.
Dependent Variable: DSSREAb.
Multiple Regression Results for ReadingCoefficientsa
1731.379 104.309 16.599 .000
-2.150 2.551 -.089 -.843 .402
-3.681 .601 -.648 -6.130 .000
(Constant)
Teacher salary in 1000sof dollars
Percentage of Studentson free or reduced lunch
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: DSSREAa.
Multiple Regression Results for Reading
What did we find with respect to H1 in the multivariate case?
Do we find support for the hypothesis that the higher the teacher salary, the better a county scored on state mandated assessment? Answer: NO! We find a very slight relationship the other way,
the higher the teacher salary the LOWER a county scored on state mandated assessment.
Multiple Regression Results for Reading
We DO find a VERY strong statistical relationship between the percentage of students in a county on free and reduced lunch and scores on state mandated assessments.
What would we conclude? At the bivariate level, with no statistical controls,
we found no relationship between teacher salary and reading performance.
Multiple Regression Results for Reading
At the multivariate level, controlling for the percentage of students on free and reduced lunch, we still find no effect.
Multiple Regression Results for Reading
Casewise Diagnosticsa
-2.705 1458.00 1609.9145 -151.9145
-2.263 1432.00 1559.1260 -127.1260
Case Number48
61
Std. Residual DSSREAPredicted
Value Residual
Dependent Variable: DSSREAa.
Residuals Statisticsa
1398.3772 1609.9146 1501.3284 43.75312 67
-151.9145 109.6565 .0000 55.31160 67
-2.353 2.482 .000 1.000 67
-2.705 1.952 .000 .985 67
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum Maximum Mean Std. Deviation N
Dependent Variable: DSSREAa.
Multiple Regression Results for MathDescriptive Statistics
1440.5970 57.05581 67
36.1378 2.92448 67
41.3897 12.42106 67
DSSMATH
Teacher salary in 1000sof dollars
Percentage of Studentson free or reduced lunch
Mean Std. Deviation N
Variables Entered/Removedb
Percentage ofStudentson free orreducedlunch,Teachersalary in1000s ofdollars
a
. Enter
Model1
VariablesEntered
VariablesRemoved Method
All requested variables entered.a.
Dependent Variable: DSSMATHb.
Multiple Regression Results for MathModel Summaryb
.639a .408 .390 44.56481Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), Percentage of Students on freeor reduced lunch, Teacher salary in 1000s of dollars
a.
Dependent Variable: DSSMATHb.
ANOVAb
87748.684 2 43874.342 22.092 .000a
127105.4 64 1986.022
214854.1 66
Regression
Residual
Total
Model1
Sum ofSquares df Mean Square F Sig.
Predictors: (Constant), Percentage of Students on free or reduced lunch, Teachersalary in 1000s of dollars
a.
Dependent Variable: DSSMATHb.
Multiple Regression Results for Math
Coefficientsa
1547.872 82.759 18.703 .000
.356 2.024 .018 .176 .861
-2.903 .476 -.632 -6.093 .000
(Constant)
Teacher salary in 1000sof dollars
Percentage of Studentson free or reduced lunch
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: DSSMATHa.
Multiple Regression Results for Math What did we find with respect to the H1 in the
multivariate case for math? Do we find support at the multivariate level for the
hypothesis that the higher the teacher salary, the better a county scored on state mandated assessment? Answer: NO! We find a very slight positive relationship, but the
effect is not close to what we need to claim “statistical significance”.
Multiple Regression Results for MathCasewise Diagnosticsa
-2.720 1418.00 1539.2003 -121.2003
-2.299 1388.00 1490.4589 -102.4589
2.453 1516.00 1406.6899 109.3101
Case Number48
61
67
Std. Residual DSSMATHPredicted
Value Residual
Dependent Variable: DSSMATHa.
Residuals Statisticsa
1350.4028 1539.2003 1440.5970 36.46266 67
-121.2003 109.3101 .0000 43.88439 67
-2.474 2.704 .000 1.000 67
-2.720 2.453 .000 .985 67
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum Maximum Mean Std. Deviation N
Dependent Variable: DSSMATHa.