Normality Assumption, 1
Model Diagnostics: Normality Assumption
Scenario
Data and Instructions
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Download the Healthy_SCH23.xlsx subset of the Healthy Minds 2021-2022 data file from D2L.
Upload the Healthy_SCH23.xlsx to SAS Studio.
Upload your SAS code to D2L.
Upload a Word document that contains ONLY the following:
 Task 2. The raw critical values for skewness and kurtosis.
 Task 3.The summary table for Flourishing.
 Task 7. The summary table for the residuals.
 Task 8. The summary table for the residuals.
 Your answers to the questions in Task 9.
 The portion of the APA results section that deals with descriptive statistics, Pearson correlations,
and normality issues. Refer to the example in the Model Diagnostics notes.
Hypothesis

Feelings of loneliness can be predicted from a combination of depression, anxiety, and positive mental health
feelings.
Variable
LoneSc
Item
Coding
Sum of loneliness variables. Higher scores indicate greater feelings of loneliness.
Depression
Sum of depression variables. Higher scores indicate greater feelings of
depression.
Sum of anxiety variables. Higher scores indicate greater feelings of anxiety.
Min = 3
Max = 9
Min = 9
Max = 36
Min = 7
Max = 28
Min = 8
Max = 56
Anxiety
Flourishing
Sum of positive mental health items. Higher scores indicate more positive
mental health feelings.
LAB 3:
LAB 1:
Check for impossible values.
Explore variable
transformations.
Evaluate normality
assumption of the initial
regression model.
Check for outliers.
LAB 2:
Evaluate linearity assumption
of the regression model.
Check for collinearity among
the predictors in the
regression model.
Evaluate homoscedasticity
assumption of the regression
model.
Re-evaluate normality
assumption of the final
regression model.
Obtain bootstrap confidence
intervals for the regression
model.
Normality Assumption, 2
Task 1. Bring in the Data and Create the UpdatedHealthy Data Set.
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Download the LoneSC – Data Management.sas file from D2L.
Upload the LoneSC – Data Management.sas file to SAS Studio.
Double-click on the LoneSC – Data Management.sas to edit it.
Press F3 or the run icon to submit the code. This will create a data set called UpdatedHealthy.
Task 2. Determine Skewness and Kurtosis Critical Values
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
Download the Standardized Skewness and Kurtosis – Updated.xlsx file from D2L.
Enter the sample size of 262 to obtain the critical values for skewness and kurtosis.
We want to determine whether skewness and kurtosis values are significant. Fortunately, the sample size is 262
for all of the variables we are analyzing. This allows us to convert the Z score critical values to critical values
for the raw scale. Once we obtain the values, we can compare all of the skewness and kurtosis values to the
same raw scale values. This approach is NOT appropriate if there are different sample sizes for each variable.
Write the raw scale critical values for skewness on the guidelines shown below.
Negatively Skewed
Symmetric
-Critical Value
0
Positively Skewed
+Critical Value
Write the raw scale critical values for kurtosis on the guidelines shown below.
Platykurtic
-Critical Value
Mesokurtic
0
Leptokurtic
+Critical Value
Normality Assumption, 3
Task 3. Evaluate the Flourishing Distribution
► Tasks and Utilities ► Tasks ►Statistics ►Distribution Analysis
Data Tab Settings
DATA: work.UpdatedHealthy
Analysis Variables: Flourishing
Options Tab Settings
CHECKING FOR NORMALITY
●Histogram and goodness-of-fit tests
● Add inset statistics
● Normal quantile-quantile plot
● Add inset statistics
INSET STATISTICS
● Number of Observations ● Goodness-of-fit test ● Mean ● Skewness ● Kurtosis
Edit the Code
● Replace GoodnessOfFit with TestsForNormality. Then, press F3 or the run icon to submit the code.
Summarize the distribution analysis for Flourishing in the table shown below. Since Flourishing does not have a normal
distribution, variable transformations should be considered.
● Add the following variables to the Flourishing – Data Management.sas program.
● Press F3 or the run icon to submit the code.
log_Flourishing = log(Flourishing);
sqr_Flourishing = Flourishing**2;
●Save the updated Flourishing_Data Management.sas program.
● Repeat the distribution analysis for Log_Flourishing and sqr_Flourishing.
Summarize the Results
● Complete the table shown below to summarize the distribution analyses you conducted.
Flourishing
Shapiro-Wilk W
Shapiro-Wilk p value
Conclusion
Skewness
Conclusion
Kurtosis
Conclusion
Which representation is best?
Log_Flourishing
Sqr_Flourishing
Normality Assumption, 4
Task 4. Evaluate the LoneSc Distribution
The output needs to determine the best representation for LoneSc has been summarized below and the graphs are
shown on the next page. You just need to review the summary table and the graphs to determine the best
representation for LoneSc. The instructions are provided here to demonstrate all of the steps involved for a real research
project.
► Tasks and Utilities ► Tasks ►Statistics ►Distribution Analysis
Data Tab Settings
DATA: work.UpdatedHealthy
Analysis Variables: LoneSc
Options Tab Settings
CHECKING FOR NORMALITY
●Histogram and goodness-of-fit tests
● Add inset statistics
● Normal quantile-quantile plot
● Add inset statistics
INSET STATISTICS
● Number of Observations ● Goodness-of-fit test ● Mean ● Skewness ● Kurtosis
Edit the Code
● Replace GoodnessOfFit with TestsForNormality. Then, press F3 or the run icon to submit the code.
The distribution analysis for LoneSc has been summarized in the table shown below. Since LoneSc does not have a
normal distribution, variable transformations should be considered.
● Add the following variables to the LoneSc – Data Management.sas program.
log_lonesc = log(lonesc);
sqr_lonesc = lonesc**2;
● Press F3 or the run icon to submit the code.
● Repeat the distribution analysis for Log_LoneSc and sqr_lonesc.
Summarize the Results
● Complete the table shown below to summarize the distribution analyses you conducted.
LoneSc
Log_LoneSc
Sqr_LoneSc
Shapiro-Wilk W
Shapiro-Wilk p value
Conclusion
.919
< .0001
Nonnormal
.903
< .0001
Nonnormal
.894
< .0001
Nonnormal
Skewness
Conclusion
.006
Symmetric
-0.48
Negatively Skewed
0.43
Positively Skewed
Kurtosis
Conclusion
-1.09
Platykurtic
-0.83
Platykurtic
-1.02
Platykurtic
Which representation is best?
Normality Assumption, 5
Normality Assumption, 6
Task 5. Evaluate the Depression Distribution
The output needs to determine the best representation for Depression has been summarized below and the graphs are
shown on the next page. You just need to review the summary table and the graphs to determine the best
representation for depression. The instructions are provided here to demonstrate all of the steps involved for a real
research project.
► Tasks and Utilities ► Tasks ►Statistics ►Distribution Analysis
Data Tab Settings
DATA: work.UpdatedHealthy
Analysis Variables: Depression
Options Tab Settings
CHECKING FOR NORMALITY
●Histogram and goodness-of-fit tests
● Add inset statistics
● Normal quantile-quantile plot
● Add inset statistics
INSET STATISTICS
● Number of Observations ● Goodness-of-fit test ● Mean ● Skewness ● Kurtosis
Edit the Code
● Replace GoodnessOfFit with TestsForNormality. Then, press F3 or the run icon to submit the code.
The distribution analysis for depression has been summarized in the table shown below. Since depression does not have
a normal distribution, variable transformations should be considered.
● Add the following variables to the Depression – Data Management.sas program.
log_Depression = log(Depression);
sqr_Depression = Depression**2;
● Press F3 or the run icon to submit the code.
● Repeat the distribution analysis for Log_Depression and sqr_Depression.
Summarize the Results
● Complete the table shown below to summarize the distribution analyses you conducted.
Shapiro-Wilk W
Shapiro-Wilk p value
Conclusion
Skewness
Conclusion
Kurtosis
Conclusion
Depression
.953
< .0001
Nonnormal
0.58
Positively skewed
-0.44
Mesokurtic
Log_Depression
.981
.0015
Nonnormal
-0.08
Symmetric
-0.72
Platykurtic
Sqr_Depression
.876
< .0001
Nonnormal
1.17
Positively skewed
0.68
Leptokurtic
Which representation is best?
I would use log depression, because the skewness is symmetric, and ketosis is only slightly platykurtic. Depression has a
positively skewed Skewness value that’s slightly more intense (not sure how else to describe it) than the platykurtic value
of log depression.
Normality Assumption, 7
Normality Assumption, 8
Task 6. Evaluate the Anxiety Distribution
The output needs to determine the best representation for anxiety has been summarized below and the graphs are
shown on the next page. You just need to review the summary table and the graphs to determine the best
representation for anxiety. The instructions are provided here to demonstrate all of the steps involved for a real research
project.
► Tasks and Utilities ► Tasks ►Statistics ►Distribution Analysis
Data Tab Settings
DATA: work.UpdatedHealthy
Analysis Variables: Anxiety
Options Tab Settings
CHECKING FOR NORMALITY
●Histogram and goodness-of-fit tests
● Add inset statistics
● Normal quantile-quantile plot
● Add inset statistics
INSET STATISTICS
● Number of Observations ● Goodness-of-fit test ● Mean ● Skewness ● Kurtosis
Edit the Code
● Replace GoodnessOfFit with TestsForNormality. Then, press F3 or the run icon to submit the code.
The distribution analysis for Anxiety has been summarized in the table shown below. Since Anxiety does not have a
normal distribution, variable transformations should be considered.
● Add the following variables to the Anxiety – Data Management.sas program. Then, press F3 or the run icon to submit
the code.
log_Anxiety = log(Anxiety);
sqr_Anxiety = Anxiety**2;
● Repeat the distribution analysis for Log_Anxiety and sqr_Anxiety.
Summarize the Results
● Complete the table shown below to summarize the distribution analyses you conducted.
Shapiro-Wilk W
Shapiro-Wilk p value
Conclusion
Skewness
Conclusion
Kurtosis
Conclusion
Anxiety
.956
< .0001
Nonnormal
0.27
Symmetric
-0.97
Platykurtic
Log_Anxiety
.955
< .0001
Nonnormal
-0.37
Negatively Skewed
-0.75
Platykurtic
Sqr_Anxiety
.903
< .0001
Nonnormal
0.79
Positively Skewed
-0.455
Mesokurtic
Which representation is best?
I would use Anxiety, not any of the transformed data because Anxiety has the less severe variables that indicate
skewness or kurtosis of the two that don’t have both non symmetrical/mesokurtic data.
Normality Assumption, 9
Normality Assumption, 10
Task 7. Evaluate Residuals of Linear Regression Model Using Original Variables
Obtain Residuals from the Regression Model:
► Tasks and Utilities ► Tasks ►Linear Models ►Linear Regression
Data Tab Settings
DATA: work.UpdatedHealthy
Dependent Variable: LoneSc
Continuous Variables: Flourishing, Depression, Anxiety
Model Tab Settings
Model Effects: Intercept, Flourishing, Depression, Anxiety
Options Tab Settings
You can use the default and just ignore the graphs or you can unselect all of the plots.
Output Tab Settings
● Create observationwise statistics data set
Data Set Name: work.Aresiduals
● Residual
Press F3 or the run icon to submit the code.
Analyze the Residuals from the Regression Model
► Tasks and Utilities ► Tasks ►Statistics ►Distribution Analysis
Data Tab Settings
DATA: work.Aresiduals
Analysis Variables: r_
Options Tab Settings
CHECKING FOR NORMALITY
●Histogram and goodness-of-fit tests
● Add inset statistics
● Normal quantile-quantile plot
● Add inset statistics
INSET STATISTICS
● Number of Observations ● Goodness-of-fit test ● Mean ● Skewness ● Kurtosis
Edit the Code
● Replace GoodnessOfFit with TestsForNormality.
Press F3 or the run icon to submit the code. Complete the table below based on the output generated.
Shapiro-Wilk W
Shapiro-Wilk p value
Conclusion
Residuals using Original Variables
.98
<.0001
Nonnormal
Skewness
Conclusion
.49
Positively skewed
Kurtosis
Conclusion
.06
Mesokurtic
Normality Assumption, 11
Task 8. Evaluate the Residuals of the Linear Regression Model Using Transformed Variables
Obtain Residuals from the Regression Model:
► Tasks and Utilities ► Tasks ►Linear Models ►Linear Regression
Data Tab Settings
Dependent Variable: LoneSc
Continuous Variables: Sqr_Flourishing, Log_Depression, Anxiety
Model Tab Settings
Model Effects: Intercept, Sqr_Flourishing, Log_Depression, Anxiety
Options Tab Settings
You can use the default and just ignore the graphs or you can unselect all of the plots.
Output Tab Settings
● Create observationwise statistics data set
Data Set Name: work.Bresiduals
● Residual
Press F3 or the run icon to submit the code.
Analyze the Residuals from the Regression Model
► Tasks and Utilities ► Tasks ►Statistics ►Distribution Analysis
Data Tab Settings
DATA: work.Bresiduals
Analysis Variables: r_
Options Tab Settings
CHECKING FOR NORMALITY
●Histogram and goodness-of-fit tests
● Add inset statistics
● Normal quantile-quantile plot
● Add inset statistics
INSET STATISTICS
● Number of Observations ● Goodness-of-fit test ● Mean ● Skewness ● Kurtosis
Edit the Code
● Replace GoodnessOfFit with TestsForNormality.
Press F3 or the run icon to submit the code. Complete the table below based on the output generated.
Shapiro-Wilk W
Shapiro-Wilk p value
Conclusion
Residuals using Transformed Variables
.99
.02
Nonnormal
Skewness
Conclusion
.45
Positively Skewed
Kurtosis
Conclusion
.35
Mesokurtic
Normality Assumption, 12
Task 9. Decide which model(s) will be used going forward (i.e., in lab 2 and lab 3).
Which model should be used going forward (i.e., in lab 2 and lab 3)? Why?
You want to use the original going forward because even after transformation, the subsequent analyses did not yield
better results in some way or another so we should stick with the original.
APA Results Section
Results
See Table 1 for descriptive statistics and Table 2 for Pearson correlations among the variables.
Skewness, kurtosis, and Shapiro-Wilk’s W tests were evaluated to determine whether variable transformations
should be considered (see Table 1). Flourishing was negatively skewed log_flourishing and sqr_florishing was
created. Depression was positively skewed, so log_depression and Sqr_depression was created. The residual
distribution from the regression model using the original scaling of all variables (Shapiro-Wilk’s W = 0.98, p
<.0001, Skewness = .49, Kurtosis = .06) and the residual distribution from the regression model including the
transformed variables (Shapiro-Wilk’s W = 0.99, p = .02, Skewness = 0.45, Kurtosis = .35) were similar, so the
original scaling of variables was used to make interpretation easier.
Table 1
Descriptive Statistics for All Variables (N = 262)
Variable
Shapiro-Wilk
Test for
Normality
W
p
M
SD
Skewness
Kurtosis
Loneliness
6.07
1.96
.01
-1.09
.92
<.0001
Flourishing
42.54
9.73
-1.13
1.05
.91
<.0001
Depression
18.82
6.84
.58
-.44
.95
<.0001
Anxiety
18.02
7.10
.27
-.97
.96
<.0001
Table 2
Pearson Correlations for All Variables (N = 262)
1
1.
Loneliness
2.
Flourishing
3.
Depression
4.
Anxiety
**p < .01.
--
2
3
4
-.61**
<.0001
--
.60**
<.0001
-.511**
<.0001
--
.52**
<.0001
-.40**
<.0001
.76**
<.0001
--
Normality Assumption, 13