Multiple Regression II

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


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


Sig. (1-tailed)

N


Sig. (1-tailed)

N

DSSMATH

DSSREA


Spearman's rhoDSSMATH DSSREA


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


Mean Std. Deviation N

Variables Entered/Removedb


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.


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.


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)


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


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


Results for simple regression: Math


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





a. Enter

Model1

VariablesEntered



Dependent Variable: DSSREAb.


Model Summaryb

.154a .024 .009 70.21537Model1



Predictors: (Constant), Teacher salary in 1000s ofdollars

a.


ANOVAb

7801.938 1 7801.938 1.582 .213a

320462.8 65 4930.198

328264.8 66

Regression

Residual

Total

Model1


Predictors: (Constant), Teacher salary in 1000s of dollarsa.



Coefficientsa

1366.977 107.144 12.758 .000

3.718 2.955 .154 1.258 .213

(Constant)


Model1

B Std. Error


Beta


t Sig.

Dependent Variable: DSSREAa.


Casewise Diagnosticsa

-2.535 1314.00 1492.0236 -178.0236Case Number33

Std. Residual DSSREAPredicted

Value Residual


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. Residual



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



Percentage ofStudentson free orreducedlunch,Teachersalary in1000s ofdollars

a

. Enter

Model1

VariablesEntered




Multiple Regression Results for ReadingModel Summaryb

.620a .385 .366 56.16920Model1



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


Predictors: (Constant), Percentage of Students on free or reduced lunch, Teachersalary in 1000s of dollars

a.


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)



Model1

B Std. Error


Beta


t Sig.


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.


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.


At the multivariate level, controlling for the percentage of students on free and reduced lunch, we still find no effect.


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



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. Residual



Multiple Regression Results for MathDescriptive Statistics

1440.5970 57.05581 67

36.1378 2.92448 67

41.3897 12.42106 67

DSSMATH





Percentage ofStudentson free orreducedlunch,Teachersalary in1000s ofdollars

a

. Enter

Model1

VariablesEntered




Multiple Regression Results for MathModel Summaryb

.639a .408 .390 44.56481Model1



Predictors: (Constant), Percentage of Students on freeor reduced lunch, Teacher salary in 1000s of dollars

a.


ANOVAb

87748.684 2 43874.342 22.092 .000a

127105.4 64 1986.022

214854.1 66

Regression

Residual

Total

Model1


Predictors: (Constant), Percentage of Students on free or reduced lunch, Teachersalary in 1000s of dollars

a.


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)



Model1

B Std. Error


Beta


t Sig.


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



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. Residual



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Multiple Regression II