Lecture 6 guidelines_and_assignment

Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines
Prof. Dr. Chang Zhu page 1
Table of Contents
LECTURE 6 .....................................................................................................................................................2
LINEAR REGRESSION .....................................................................................................................................2
MULTIPLE REGRESSION.................................................................................................................................3
SPSS OUTPUT ............................................................................................................................................5
HYPOTHESIS TESTING................................................................................................................................5
REPORT THE RESULTS ...............................................................................................................................8
MODEL GENERALIZATION.........................................................................................................................9
ASSIGNMENT 6............................................................................................................................................13
References ..................................................................................................................................................13

LECTURE 6
LINEAR REGRESSION
We use linear regression when we want to know the relationship between 2 variables. One of them is
call the independent (predictor) variable and the other dependent (outcome) variable.
e.g. we want to know if the time spent to study Statistics will predict one’s score in the course.
The data file is named study_time.sav
In SPSS, choose Analyse > Regression > Linear
Move the Exam_scores to the Dependent box and the Hours variable to the Independent(s) box.
Click OK to run the analysis.
In the output, we should look first at the Model Summary table. In linear regression, the R value (R
= .827) is simply the Pearson correlation coefficient between two variables. The R square (R2
= .684) is
used in percentage to inteprete the variation in the outcome variable (the exam score): It shows that the
number of hours spent on studying accounts for 68.4% of variation in the exam scores.
Model Summaryb
Model R R Square Adjusted R Square
Std. Error of the
Estimate
1 .827a
.684 .666 9.411
a. Predictors: (Constant), Hours
b. Dependent Variable: Exam_scores
The F-statistic that tests whether the model has improved the prediction of the outcome in compared to
one that uses the mean as the predicted value, which can be found in the ANOVA table, F = 38.959, p
< .01.
ANOVAa
Model Sum of Squares df Mean Square F Sig.
1 Regression 3450.712 1 3450.712 38.959 .000b
Residual 1594.308 18 88.573
Total 5045.020 19
a. Dependent Variable: Exam_scores
b. Predictors: (Constant), Hours

The model parameters can be found in the Coefficients table. The constant (b0) is to mean that when no
hours is spent on studyding, the predicted exam score would be 12.762. The next b value (usually called
b1 in the equation) is 2.391 can be explaied such that for one unit of change (one hour) in the number of
hours studying, the model predicts that an increase of 2.391 in the exam score will be observed.
We can report the result as follows:
A linear regression analysis revealed that the number of hours spent studying was a highly significant
predictor of exam scores ( = .2.391, p = < .01), accounting for 68.4% of the variance in exam scores.
MULTIPLE REGRESSION
This example is taken from a real research study conducted by (Ong, Ho, Lim, Goh, Lee, & Chua, 2011).
The study can be summarized as follow:
The authors want to examine the relationship between narcissism (“characterized by a highly inflated,
positive but unrealistic self-concept, a lack of interest in forming strong interpersonal relationships, and
an engagement in self-regulatory strategies to affirm the positive self-views”, Campbell & Foster, 2007
cited in Ong et al., 2011) and behavior on Facebook in adolescents from grade 7 to 9 (n= 275). They
measured the Age, Gender, and Grade (at school), extraversion and narcissism.
Extraversion is measured by using the 12-item Extraversion subscale of the NEO Five-Factor Inventory
(NEO-FFI, Costa & McCrae, 1992a)
Narcissism is measured by 12-item Narcissistic Personality Questionnaire for Children-Revised (NPQC-
R, Ang & Raine, 2009).
The outcome variable is named FB_status, which is measured by how often the participants update their
status per week.
They hypothesized that narcissism would predict, above and beyond the other variables, the frequency
of status updates.
The data file is named narcissism.sav, which is adapted from Field (2013).
Coefficients
a
Model
Unstandardized Coefficients
Standardized
Coefficients
t Sig.
95.0% Confidence Interval for B
B Std. Error Beta Lower Bound Upper Bound
1 (Constant) 12.762 8.276 1.542 .140 -4.625 30.150
Hours 2.391 .383 .827 6.242 .000 1.586 3.196
a. Dependent Variable: Exam_scores

Observation:
The study’s hypothesis is that:
H1: Narcissism will predict a higher frequency of updating Facebook status over and above extraversion.
which means the authors want to find out if narcisissism can explain more unique variation in the
freqency of Facebook status update among adolescents, after controlling for demographic information
(age, gender, and grade), and extraversion.
Therefore, we will opt for a hierarchical regression where age, gender, and grade will be entered in the
first block, extraversion in the second block, and finally, narcissism in the third block.
In SPSS, choose Analyse > Regression > Linear
First move the outcome variable FB_Status to the Dependent box.
The first block of Independent(s) variables will consist of Grade, Age, and Gender. Click to access
the second block so as to enter the Extraversion variable.
After that, click Next to access block 3, and move the variable Narcissism to the Independent(s) box.
Then click on the Statistics button to select the analysis options as below.
Click on Continue to proceed to the next step.
The options offered by SPSS in the Plots and Save tabs are mainly for assumptions testing.
In the Plots: when we plot the ZRESID (standardized residuals) on the Y-axis against the ZPRED
(standardized predicted values) on the X-axis, we can determine whether the assumptions of random
errors and homoscedascity have been met. So we will move the ZPRED and the ZRESID to the X-axis and
the Y-axis, respectively as illustrated.
Field (2009) also suggests to plot the SRESID (Studentized residuals) on the Y-axis against the ZPRED
(standardized predicted values) on the X-axis because the graph will be more case-by-case basis. To do
this, we click Next, and move the ZPRED and the SRESID to the X-axis and the Y-axis accordingly.
Click Continue to proceed to the next step.

The dialog box named Save (Saving regression analysis) helps us to evaluate how well our model fits the
data and to detect any cases that have an influence on the model. Select the options as suggested, then
click Continue to proceed to the next step.
The Option dialog box provides us with the option to choose the probability level for our analysis as well
as whether to include a constant in the regression equation. For missing values, the recommended
option is Exclude cases listwise.
SPSS OUTPUT
HYPOTHESIS TESTING
As with linear regression, we will look at 3 tables: Model Summary, Anova, and Coefficients to come up
with the conclusion.
As we entered the variables in 3 blocks with the first block being the variables that have already been
confirmed, and the final (third) block being the variable (Narcissism) of our interest, we will find 3
models in the Model Summary table:
 Model 1 includes Gender, Age, and Grade as predictors, and accounts for 4% in the variation in
Facebook status update.
 Model 2 includes Gender, Age, and Grade, and Extraversion, and accounts for 5.6% in the
variation (R squared change indicated as ΔR2
is equal to 1.6%).

 Similarly, model 3 includes Gender, Age, and Grade, Extraversion, and Narcissism and accounts
for 9% (ΔR2
= 3.4%).
The Durbin-Watson statistic is 2.032, indicating that the assumption of independent errors is met.
Model Summaryd
Model R
R
Square
Adjusted R
Square
Std. Error of the
Estimate
Change Statistics
Durbin-
Watson
R Square
Change
F
Change df1 df2
Sig. F
Change
1 .200a
.040 .028 2.45090 .040 3.426 3 247 .018
2 .236b
.056 .040 2.43550 .016 4.133 1 246 .043
3 .299c
.090 .071 2.39648 .034 9.078 1 245 .003 2.032
a. Predictors: (Constant), Gender, Age, Grade
b. Predictors: (Constant), Gender, Age, Grade, Extraversion
c. Predictors: (Constant), Gender, Age, Grade, Extraversion, Narcissism
d. Dependent Variable: Frequency of changing status per week
The ANOVA table shows that all the three models significantly improve our ability to predict the
outcome (Facebook status update) compared to a model based just on the mean.
ANOVAa
Model Sum of Squares df Mean Square F Sig.
1 Regression 61.732 3 20.577 3.426 .018b
Residual 1483.712 247 6.007
Total 1545.444 250
2 Regression 86.250 4 21.563 3.635 .007c
Residual 1459.194 246 5.932
Total 1545.444 250
3 Regression 138.384 5 27.677 4.819 .000d
Residual 1407.061 245 5.743
Total 1545.444 250
a. Dependent Variable: Frequency of changing status per week
b. Predictors: (Constant), Gender, Age, Grade
c. Predictors: (Constant), Gender, Age, Grade, Extraversion
d. Predictors: (Constant), Gender, Age, Grade, Extraversion, Narcissism
The contribution of each individual to the regression model when others are held constant can be found
in the Coeficients table.
The Durbin-Watson indicates
independent of errors,
should be greater than 1 and
less than 3

As can be seen from the Coefficients table, the VIF and the tolerance statistics for each predictor are at
acceptable cut-off values, therefore, assumption of no multicollinearity is met. However, if we look at
the Collinearity Diagnostics table, we see Grade and Age both have high loadings in the sixth dimension.
This is due to the fact the ages of students in a certain grade can be quite similar.
Collinearity Diagnosticsa
Model Dimension Eigenvalue Condition Index
Variance Proportions
(Constant) Grade Age Gender Extraversion Narcissism
1 1 3.369 1.000 .00 .00 .00 .03
2 .562 2.449 .00 .01 .00 .83
3 .068 7.025 .01 .26 .00 .12
4 .001 68.541 .99 .73 1.00 .02
2 1 4.324 1.000 .00 .00 .00 .01 .00
2 .578 2.735 .00 .00 .00 .84 .00
3 .085 7.121 .00 .24 .00 .08 .04
4 .012 18.915 .03 .04 .02 .05 .93
5 .001 78.759 .97 .72 .98 .01 .03
3 1 5.276 1.000 .00 .00 .00 .01 .00 .00
2 .581 3.013 .00 .00 .00 .80 .00 .00
3 .097 7.376 .00 .21 .00 .09 .01 .06
4 .034 12.398 .01 .03 .00 .00 .01 .76
5 .011 22.276 .02 .03 .01 .09 .95 .17
6 .001 87.003 .97 .72 .98 .01 .02 .00
Coefficientsa
Model
Unstandardized
Coefficients
Standardized
Coefficients
t Sig.
95.0% Confidence Interval for B Correlations Collinearity Statistics
B Std. Error Beta Lower Bound Upper Bound Zero-order Partial Part Tolerance VIF
1 (Constant) 3.383 3.674 .921 .358 -3.852 10.619
Grade -.444 .388 -.149 -1.145 .253 -1.208 .320 -.131 -.073 -.071 .229 4.365
Age -.033 .309 -.014 -.107 .915 -.642 .576 -.129 -.007 -.007 .236 4.233
Gender -.775 .327 -.153 -2.370 .019 -1.420 -.131 -.122 -.149 -.148 .936 1.068
2 (Constant) .830 3.861 .215 .830 -6.775 8.434
Grade -.486 .386 -.163 -1.259 .209 -1.246 .274 -.131 -.080 -.078 .228 4.378
Age -.006 .308 -.002 -.019 .985 -.612 .600 -.129 -.001 -.001 .236 4.241
Gender -.691 .328 -.136 -2.110 .036 -1.337 -.046 -.122 -.133 -.131 .921 1.085
Extraversion .052 .025 .127 2.033 .043 .002 .101 .137 .129 .126 .977 1.024
3 (Constant) .650 3.799 .171 .864 -6.833 8.134
Grade -.522 .380 -.175 -1.375 .170 -1.271 .226 -.131 -.087 -.084 .228 4.382
Age -.010 .303 -.004 -.033 .974 -.606 .586 -.129 -.002 -.002 .236 4.241
Gender -.943 .333 -.186 -2.831 .005 -1.599 -.287 -.122 -.178 -.173 .864 1.158
Extraversion .011 .028 .028 .394 .694 -.045 .067 .137 .025 .024 .758 1.320
Narcissism .066 .022 .212 3.013 .003 .023 .110 .187 .189 .184 .752 1.329
The VIF and the tolerance statistics helps
to evaluate the assumption of no
multicollinearity with VIF greater than 10
and tolerance less than .2 being causes of
concern.

If we look at model 3, i.e. after controlling for grade, age, gender, it’s found that narcissism significantly
predicts the frequency of Facebook status update over and above/beyond extraversion (β = .21, p <.01)
REPORT THE RESULTS
Field (2009) suggests that we should report the the constant, the unstandardized betas and their
standard errors, the standardized betas with their significance level indicated in the footnote as well as
the R square change (ΔR2
) for each step of the analysis.
We can write:
A hierachical multiple regression was conducted to examine if alfter controlling for grade, gender, and
age, narcissism can significantly predict the frequency of Facebook status update among adolescences
over and above extraversion. The result confirms the hypothesis such that narcissism accounted for a
significant variance in the frequency of Facebook status update over and above extraversion, ΔR2
= .03,
ΔF(1, 245) = 9.08, p < .01).
The results can be found in Table 1.
Table 1
Summary Of Hierarchical Multiple Regression Analyses For Extraversion And Narcissism Predicting The
Frequency Of Facebook Status Updates
B Std. Error Beta
Step 1
(Constant) 3.383 3.674
Grade -.444 .388 -.149
Age -.033 .309 -.014
Gender -.775 .327 -.153*
Step 2
(Constant) .830 3.861
Grade -.486 .386 -.163
Age -.006 .308 -.002
Gender -.691 .328 -.136*
Extraversion .052 .025 .127*
Step 3
(Constant) .650 3.799
Grade -.522 .380 -.175
Age -.010 .303 -.004
Gender -.943 .333 -.186*
Extraversion .011 .028 .028
Narcissism .066 .022 .212**
Notes R2
= .04 for Step 1, ΔR2
= .016 for Step 2 (p < .05), ΔR2
= .034 for Step 3 (p < .01). *
p<.05, **
p<.01

MODEL GENERALIZATION
To evaluate whether we generalize our model to make generalization about a different
sample/population we should look at the following graphs that we have requested in the analysis.
The first two are the histogram and the normal probability plot with the standardized predicted values
(ZPRED) against the standardized residuals (ZRESID). As can be seen, the histogram and the P-P plot
show that deviation from normality has been found.
The scatter plots also indicate that there is problem of heteroscedasticity in the data. Field notes that “In
a situation in which the assumptions of linearity and homoscedasticity are met, the points are randomly
and evenly dispersed throughout the plot” (p. 247). So the scatter plots obtained from the study’s data
demonstrate that the distribution of residuals are not random, but follow almost a certain linear pattern.
So the assumption of homoscedasticity has not been met.

Next comes the 5 partial scatterplots of the residuals of the outcome variable (FB status update) and
each of the five predictors (grade, age, gender, extraversion, and narcissism). We can also identify any
outliers if any in the partial scatterplots. Here we just look at 2 partial scatter plots as an example, and
we notice: (1) there a linear relationship between the gender, narcissism and the Facebook update
status; (2) there are 2 outliers.
Actually, we can already identify which are the outliers in the table named Casewise Diagnostics. In this
case, they are case 131 and 231 because we can see that their Facebook status update per week is 14
while the model predicts just 2.31 (residual = 11.68) and 2.26 (residual = 11.73) accordingly.
Casewise Diagnostics
a
Case Number Std. Residual
Frequency of
changing status
per week Predicted Value Residual
131 4.878 14.00 2.3104 11.68955
231 4.896 14.00 2.2676 11.73244
To see if these two cases will have significant influence on the accuracy of our regression model we will
examine if they exceed the conventional cut-off values of the following influential statistics (Field, 2009):
 Calculate the average leverage (number of predictor plus 1, divided by the sample size or
(k+1)/n), look for value greater than two or three times this average value. So for our data, the
leverage value is (5+1)/251=0.024
 Cook’s distance: value above 1 indicates an influencing case.
 Mahalanobis distance, value above 15 (sample size = 100) is cause of concern.
 Absolute DFbeta greater than 1 is a problem.
 Standardized DFFit as close to zero indicates good fit.

 Calculate the upper and lower limit of acceptable values for the covariance ratio (CVR), using the
following equations. Any cases outside these limits are a problem.
upper limit for CVR: 1 + 3(k+1)/n = 1 + 3(5+1)/251 = 1.071
lower limit for CVR: 1 - 3(k+1)/n = 1 - 3(5+1)/251 = 0.928
Now we will use the Select cases command to select cases with standardized predicted residuals greater
than 3.
In SPSS, Data > Select Cases
Then choose If condition is satisfied, click If to provide the criterion.
We will move the variable ZRE_1 (Standardized Residual) to the condition area, then provide the
criterion that this value is greater than 3.
Click Continue to proceed and OK to finish. In the data view, we will see only cases 131 and 231 are
selected.
Then we will use the Case summaries command to look at the influential statistic values for these two
cases.
In SPSS, Analyze > Reports > Case Summaries
Move all the influential statistics (MAH_1, COO_1, LEV_1, COV_1, SDF_1, SDB0_1, SDB0=1_1, SDB2_1.
SDB3_1, SDB4_1, SDB5_1) into the Variables area.

Under the Display cases area, choose the options as suggested, then click OK to finish. In the output, we
can compare the influential statistics for the two cases with the cut-off values.
As obtained from the Case Summaries table, the two cases satisfy most of the criteria, except for the
Covariance Ratio and the standardized DFFit. Therefore, we can keep these two cases because according
to Field (2009), the Cook’s and the Mahalanobis distance for the two cases are acceptable, indicating
that they can cause a little, but not big influence on the regression model.
Case Summariesa
cases
Mahalanobis
Distance
Cook's
Distance
Centered
Leverage
Value COVRATIO
Standardized
DFFIT
Standardized
DFBETA
Intercept
Standardized
DFBETA
Grade
Standardized
DFBETA
Age
Standardized
DFBETA
Gender
Standardized
DFBETA
Extraversion
Standardized
DFBETA
Narcissism
131 3.99529 .08243 .01598 .55911 .73942 -.13512 .14031 -.01917 -.18263 .32825 .21441
231 1.53312 .04124 .00613 .55452 .52294 -.07824 -.09206 .03514 -.23134 .26096 -.02360
Total N 2 2 2 2 2 2 2 2 2 2 2
a. Limited to first 100 cases.
Conclusion: Using a number of diagnostic statistics to check accuracy of the model (the Durbin-Watson,
the VIF, the tolerance, the histogram, P-P plot, and scatter plots) and the influential cases, we can see
that the model fails to meet certain assumptions, especially the normal distribution of residuals and the
homoscedasticity. Therefore, we cannot generalize our model for the population. This is in accordance
with the limitations that Ong, Ho, Lim, Goh, Lee, and Chua (2011) have indicated in their paper.

ASSIGNMENT 6
(You can work alone or in group for this assignment. If you work in group, please stay in the same group
of previous assignments and indicate the group members in the submission document).
We know from the Collinearity Diagnostics that age and grade are highly correlated so, it can be
redundant to include both in the regression model and also can affect the accuracy of the model.
Therefore, we will only retain Age in this assignment and see if there can be any improvement in the
regression model.
For this assignment, you’ll try to test the following hypothesis:
H1: Narcissism will predict a higher rating of one’s own profile photo over and above extraversion.
which means you will find out if narcisissism can explain more unique variation in rating of Facebook
profile photo (profile_photo_rating) among adolescents, after controlling for demographic information
(age and gender), and extraversion.
When reporting the results, you should include the following:
- A paragraph describe what method of regression you used, the variables included, which
hypothesis to test and whether it is supported with the R square and F change and the
significant p value.
- A table of report includes the constant, the unstandardized betas and their standard errors,
the standardized betas with their significance level indicated in the footnote as well as the R
square change (ΔR2
) for each step of the analysis.
- Check if your model can be generalized by:
 looking at the histogram and the normal probability plot with the standardized
predicted values (ZPRED) against the standardized residuals (ZRESID), and the
scatter plots;
 analyzing if there are any influential outliers (criteria: value greater than 3 standard
deviations, see how to obtain this on page 5). If you remove the outlier from the
data set, re-run the analysis.
- Come up with a conclusion paragraph to see if the model can be generalized.
The data file is named narcissism.sav.
References
Field, A. (2009). Discovering statistics using SPSS. Sage publications.
Field, A. (2013). Discovering statistics using IBM SPSS Statistics. Sage publications
Ong, E. Y., Ho, J., Lim, J. C., Goh, D. H., Lee, C. S., & Chua, A. Y. (2011). Narcissism, extraversion and
adolescents’ self-presentation on Facebook. Personality and Individual Differences, 50(2), 180-
185.

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