Download Multiple Regression 2

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 Analysis
Statistics
N
Mean
Median
Std. Deviation
Skewness
Std. Error of Skewness
Kurtos is
Std. Error of Kurtosis
Percentiles
Valid
Missing
25
50
75
DSSMATH
67
18
1440.5970
1443.0000
57.05581
-.344
.293
.378
.578
1404.0000
1443.0000
1473.0000
DSSREA
67
18
1501.3284
1505.0000
70.52453
-.302
.293
-.128
.578
1458.0000
1505.0000
1548.0000
Teacher
salary in
1000s of
dollars
67
18
36.1378
35.8490
2.92448
.902
.293
1.258
.578
34.3820
35.8490
37.0620
FRL
67
18
41.3897
41.9300
12.42106
-.042
.293
.318
.578
33.7900
41.9300
50.9600
Univariate Analysis
DSSMATH
12
10
8
6
4
Std. Dev = 57.06
2
Mean = 1440.6
N = 67.00
0
1280.0
1320.0
1300.0
1360.0
1340.0
DSSMA TH
1400.0
1380.0
1440.0
1420.0
1480.0
1460.0
1520.0
1500.0
1540.0
Univariate Analysis
DSSREA
12
10
8
6
4
N = 67.00
.0
40
16 .0
20
16 .0
00
16 0 .0
8
15 0 .0
6
15 .0
40
15 0 .0
2
15 0 .0
0
15 .0
80
14 0 .0
6
14 0 .0
4
14 .0
20
14 0 .0
0
14 0 .0
8
13 0 .0
6
13 .0
40
13 0 .0
2
13
0
Std. Dev = 70.52
2
Mean = 1501.3
DSSREA
Univariate Analysis
Teacher salary in 1000s of dollars
14
12
10
8
6
4
Std. Dev = 2.92
2
Mean = 36.1
N = 67.00
0
30.0
32.0
31.0
34.0
33.0
36.0
35.0
38.0
37.0
40.0
39.0
42.0
41.0
Teac her s alary in 1000s of dollars
44.0
43.0
45.0
Univariate Analysis
Percentage of Students on free or reduced lunch
20
10
Std. Dev = 12.42
Mean = 41.4
N = 67.00
0
10.0
20.0
15.0
30.0
25.0
40.0
35.0
50.0
45.0
60.0
55.0
70.0
65.0
Percentage of Students on f r ee or reduced lunch
Univariate Analysis

I would conclude that all of my variables
are at least “reasonably” normally
distributed.
Pearson Product Moment Correlations
on the data
Correlations
DSSMATH
DSSREA
Teacher salary in
1000s of dollars
Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N
Pearson Correlation
Sig. (1-tailed)
N
DSSMATH
1
.
67
.929**
.000
67
.255*
.018
67
**. Correlation is s igni ficant at the 0.01 level (1-tailed).
*. Correlation is s igni ficant at the 0.05 level (1-tailed).

Did we find support for H1?
DSSREA
.929**
.000
67
1
.
67
.154
.106
67
Teacher
salary in
1000s of
dollars
.255*
.018
67
.154
.106
67
1
.
67
Spearman’s rho correlations on the
data
Correlations
Spearman's rho
DSSMATH
DSSREA
Teacher salary in
1000s of dollars
Correlation Coefficient
Sig. (1-tailed)
N
Correlation Coefficient
Sig. (1-tailed)
N
Correlation Coefficient
Sig. (1-tailed)
N
**. Correlation is s ignificant at the .01 level (1-tailed).
Teacher
salary in
1000s of
DSSMATH
DSSREA
dollars
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
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: Math
Descriptive Statistics
DSSMATH
Teacher salary i n
1000s of dollars
Mean
1440.5970
Std. Deviation
57.05581
36.1378
2.92448
N
67
67
Va riables Ente red/Remove db
Mo del
1
Va riab les
En tere d
Te ache r
sa lary in
10 00s of
a
do llars
Va riab les
Re moved
Me thod
.
a. All requ ested va riab les ente red.
b. De pen den t Variable : DSSMATH
En ter
Results for simple regression: Math
b
Mo del Su mm ary
Model
1
R
.255a
R Square
.065
Adjust ed
R Square
.051
St d. E rror of
the Es timate
55.58546
a. Predic tors: (Constant), Teacher sal ary i n 1000s of
dollars
b. Dependent Variable: DSSM ATH
ANOVAb
Model
1
Sum of
Squares
Regression 14020.806
Residual
200833.3
Total
214854.1
df
1
65
66
Mean Square
14020.806
3089.743
a. Predictors: (Constant), Teacher salary in 1000s of dollars
b. Dependent Variable: DSSMATH
F
4.538
Sig.
.037a
Results for simple regression: Math
Coefficientsa
Model
1
(Constant)
Teacher salary in
1000s of dollars
Unstandardized
Coefficients
B
Std. Error
1260.491
84.820
4.984
2.340
Standardized
Coefficients
Beta
.255
t
14.861
Sig.
.000
2.130
.037
a. Dependent Variable: DSSMATH

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.
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: Math
a
Ca sew ise Dia gno stics
Case Num ber
22
33
40
St d. Residual
-2. 340
-2. 593
-2. 388
DS SM ATH
1312.00
1284.00
1296.00
a. Dependent Variable: DSS MATH
Predic ted
Value
1442.0882
1428.1234
1428.7215
Residual
-130.0882
-144.1234
-132.7215
Results for simple regression: Math
Residuals Statisticsa
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum
Maximum
Mean
1411.7365 1484.5854 1440.5970
-144.1234
99.9684
.0000
-1.980
3.018
.000
-2.593
1.798
.000
a. Dependent Variable: DSSMATH
Std. Deviation
14.57520
55.16275
1.000
.992
N
67
67
67
67
Results for simple regression: Math
1600
1500
1400
1300
1200
30
32
34
36
38
Teac her s alary in 1000s of dollars
40
42
44
46
Simple Regression Results for
Reading
Descriptive Statistics
DSSREA
Teacher salary in
1000s of dollars
Mean
1501.3284
Std. Deviation
70.52453
36.1378
2.92448
N
Va riables Ente red/Remove db
Mo del
1
Va riab les
En tere d
Te ache r
sa lary in
10 00s of
a
do llars
Va riab les
Re moved
Me thod
.
a. All requ ested va riab les ente red.
b. De pen den t Var iable : DSSREA
En ter
67
67
Simple Regression Results for
Reading
b
Mo del Su mm ary
Model
1
R
.154a
R Square
.024
Adjust ed
R Square
.009
St d. E rror of
the Es timate
70.21537
a. Predic tors: (Constant), Teacher sal ary i n 1000s of
dollars
b. Dependent Variable: DSSREA
ANOVAb
Model
1
Regres sion
Residual
Total
Sum of
Squares
7801.938
320462.8
328264.8
df
1
65
66
Mean Square
7801.938
4930.198
a. Predic tors : (Constant), Teacher salary in 1000s of dollars
b. Dependent Variable: DSSREA
F
1.582
Sig.
.213a
Simple Regression Results for
Reading
Coefficientsa
Model
1
(Constant)
Teacher salary in
1000s of dollars
Unstandardized
Coefficients
B
Std. Error
1366.977
107.144
3.718
a. Dependent Variable: DSSREA
2.955
Standardized
Coefficients
Beta
.154
t
12.758
Sig.
.000
1.258
.213
Simple Regression Results for
Reading
Casewise Diagnosticsa
Case Number
33
Std. Residual
-2.535
DSSREA
1314.00
Predicted
Value
1492.0236
Residual
-178.0236
a. Dependent Variable: DSSREA
Residuals Statisticsa
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum
1479.7996
-178.0235
-1.980
-2.535
a. Dependent Variable: DSSREA
Maximum
1534.1420
140.3475
3.018
1.999
Mean
1501.3284
.0000
.000
.000
Std. Deviation
10.87250
69.68140
1.000
.992
N
67
67
67
67
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=.106
WE FAIL TO REJECT THE NULL FOR READING!

Multiple Regression Results for Reading
De scri ptive Statistics
DS SRE A
Teacher salary in 1000s
of dollars
Percentage of S tudents
on free or reduc ed lunch
Mean
1501.3284
St d. Deviat ion
70.52453
N
36.1378
2.92448
67
41.3897
12.42106
67
Va riables Ente red/Remove db
Mo del
1
Va riab les
En tere d
Pe rcen tag
e of
Stu den ts
on free or
red uce d
lun ch,
Te ache r
s a lary in
10 00s a
of
do llars
Va riab les
Re moved
Me thod
.
a. All requ es ted va riab les ente red.
b. De pen den t Var iable : DSSREA
En ter
67
Multiple Regression Results for Reading
b
Mo del Su mm ary
Model
1
R
.620a
R Square
.385
Adjust ed
R Square
.366
St d. E rror of
the Es timate
56.16920
a. Predic tors: (Constant), Percent age of S tudents on free
or reduced lunc h, Teac her salary in 1000s of dollars
b. Dependent Variable: DSSREA
ANOVAb
Model
1
Regres sion
Residual
Total
Sum of
Squares
126346.1
201918.6
328264.8
df
2
64
66
Mean Square
63173.067
3154.979
F
20.023
Sig.
.000a
a. Predictors: (Constant), Percentage of Students on free or reduced lunch, Teacher
salary in 1000s of dollars
b. Dependent Variable: DSSREA
Multiple Regression Results for Reading
Coefficientsa
Model
1
(Constant)
Teacher salary in 1000s
of dollars
Percentage of Students
on free or reduced lunch
a. Dependent Variable: DSSREA
Unstandardized
Coefficients
B
Std. Error
1731.379
104.309
Standardized
Coefficients
Beta
t
16.599
Sig.
.000
-2.150
2.551
-.089
-.843
.402
-3.681
.601
-.648
-6.130
.000
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
Case Number
48
61
Std. Residual
-2.705
-2.263
DSSREA
1458.00
1432.00
Predicted
Value
1609.9145
1559.1260
Residual
-151.9145
-127.1260
a. Dependent Variable: DSSREA
Residuals Statisticsa
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Minimum
1398.3772
-151.9145
-2.353
-2.705
a. Dependent Variable: DSSREA
Maximum
1609.9146
109.6565
2.482
1.952
Mean
1501.3284
.0000
.000
.000
Std. Deviation
43.75312
55.31160
1.000
.985
N
67
67
67
67
Multiple Regression Results for Math
De scri ptive Statistics
DS SM ATH
Teacher salary in 1000s
of dollars
Percentage of S tudents
on free or reduc ed l unch
Mean
1440.5970
St d. Deviat ion
57.05581
36.1378
2.92448
67
41.3897
12.42106
67
Va riables Ente red/Remove db
Mo del
1
Va riab les
En tere d
Pe rcen tag
e of
Stu den ts
on free or
red uce d
lun ch,
Te ache r
s a lary in
10 00s a
of
do llars
Va riab les
Re moved
a. All requ es ted va riab les
Me thod
.
ente red.
b. De pen den t Var iable : DSSMATH
En ter
N
67
Multiple Regression Results for Math
b
Mo del Su mm ary
Model
1
R
.639a
R Square
.408
Adjust ed
R Square
.390
St d. E rror of
the Es timate
44.56481
a. Predic tors: (Constant), Percent age of S tudents on free
or reduced lunc h, Teac her salary in 1000s of dollars
b. Dependent Variable: DSSM ATH
ANOVAb
Model
1
Regres sion
Residual
Total
Sum of
Squares
87748.684
127105.4
214854.1
df
2
64
66
Mean Square
43874.342
1986.022
F
22.092
Sig.
.000a
a. Predictors: (Constant), Percentage of Students on free or reduced lunch, Teacher
salary in 1000s of dollars
b. Dependent Variable: DSSMATH
Multiple Regression Results for Math
Coefficientsa
Model
1
(Constant)
Teacher salary in 1000s
of dollars
Percentage of Students
on free or reduced lunc h
Unstandardized
Coeffic ients
B
Std. Error
1547.872
82.759
a. Dependent Variable: DSSMATH
Standardiz ed
Coeffic ients
Beta
t
18.703
Sig.
.000
.356
2.024
.018
.176
.861
-2.903
.476
-.632
-6.093
.000
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 Math
a
Ca sew ise Dia gnostics
Case Num ber
48
61
67
St d. Residual
-2. 720
-2. 299
2.453
DS SM ATH
1418.00
1388.00
1516.00
Predic ted
Value
1539.2003
1490.4589
1406.6899
Residual
-121.2003
-102.4589
109.3101
a. Dependent Variable: DSS MATH
Residuals Statisticsa
Predicted Value
Residual
Std. Predicted Value
Std. Residual
Mi nimum
1350.4028
-121.2003
-2.474
-2.720
Maxim um
1539.2003
109.3101
2.704
2.453
a. Dependent Variable: DSSMATH
Mean
1440.5970
.0000
.000
.000
Std. Deviation
36.46266
43.88439
1.000
.985
N
67
67
67
67