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DISCOVERING STATISTICS USING SAS
ANDY FIELD
JEREMY MILES
2010
SAGE
753pages
This companion is designed for researchers who wish to perform linear models anal-
ysis using the definitional formulas. It can serve as a supplemental text for students
in a college-level theoretical or applied linear models course using The SAS System
(SAS) for computations. This book can also serve as the principal text for a special
topics course in a graduate program in statistics with linear models theory and appli-
cations courses as prerequisites. It may also serve as a useful reference for statistical
analysts who wish to use the SAS/IML product to work out the formulas for new
experimental methods of data analysis.
A linear models course can be taught from different approaches. Some of these
approaches are more theoretical and focus on computation and derivation of the
linear algebra formulations. Such a course might consist of theorem, proof, theorem,
proof, theorem, proof, theorem, lemma, proof, etc. Some theoretical approaches
focus on geometric interpretations of the linear model using projections. Yet other
approachesaremoreappliedandfocusonthecomputer-aidedappliedanalysisusing
high-level computational procedures such as PROC GLM, MIXED, REG, etc. in
SAS, with little time spent with the analytic formulas.
This companion illustrates the theoretical linear algebra approach to teaching
linear model analysis. This companion does not teach linear models concepts, but
demonstrates how SAS/IML can be used to evaluate numerical linear model prob-
lems. It is assumed that the reader does not have any experience using SAS/IML,
but is familiar with DATA steps and basic PROC steps in the SAS system.
All the SAS examples given in this companion are self-contained, and may be
executed as written, without additional programming. In most cases, there are other
SAS procedures that are more appropriate to use in the analysis of linear models
than the IML procedure. In many cases, the analysis will be performed in both IML
and another, more appropriate SAS procedure. This is done for the following reason:
The IML approach is directed at leaning and applying the linear models formulas.
The more “canned” procedures like REG and GLM do not allow students to see the
connection between formulas and the procedures. However, because REG and GLM
are more efficient and numerically accurate than SAS/IML in many applications of
the linear model, they will also be briefly demonstrated. The canned procedures
(such as REG and GLM) are considered the standard for analysis and are provided
to demonstrate that the results obtained using the IML procedures are the same as
those using the canned procedures. This allows students to go from written analytic
formula in say mixed models analysis to IML implementation of that same analytic
formula to high level analysis using PROC MIXED. Adding the step of PROC IML
implementation provides the missing link in the learning process.
Thetopicsselectedforthiscompanion arethetopicstheauthorfoundtobeuseful
in the academic learning of the analysis of linear models. There are many additional
topics useful in the study of linear models that were not included. However, this
should give the reader a complete treatise for a course on the subject.
Exercises have been included to aid the learner. Some exercises were developed
by withholding IML implementation steps for the reader to work through. Other
exercises are developed for perhaps repetitive application of concepts introduced in
the discussion.
The examples presented take advantage of the most recent advances in SAS/IML
and are current as of SAS version 9.2. However, most of the examples will work in
earlier versions of SAS.

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关键词：Discovering Statistics statistic Discover Covering principal designed course

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如何假？
Preface
xix
How to use this book
xxiii
Acknowledgements
xxvii
Dedication
xxix
Symbols used in this book
xxx
Some maths revision
xxxii
1
Why is my evil lecturer forcing me to learn statistics?
1

1.1.
What will this chapter tell me? 1
1

1.2.
What the hell am I doing here? I don’t belong here 1
2


1.2.1. The research process 1
3

1.3.
Initial observation: finding something that needs explaining 1
3

1.4.
Generating theories and testing them 1
4

1.5.
Data collection 1: what to measure 1
7


1.5.1.
Variables 1
7


1.5.2.
Measurement error 1
10


1.5.3. Validity and reliability 1
11

1.6.
Data collection 2: how to measure 1
12


1.6.1. Correlational research methods 1
12


1.6.2. Experimental research methods 1
13


1.6.3. Randomization 1
17

1.7.
Analysing data 1
18


1.7.1. Frequency distributions 1
18


1.7.2. The centre of a distribution 1
20


1.7.3. The dispersion in a distribution 1
23


1.7.4. Using a frequency distribution to go beyond the data 1
24


1.7.5. Fitting statistical models to the data 1
26


What have I discovered about statistics? 1
28


Key terms that I’ve discovered
28


Smart Alex’s tasks
29


Further reading
29


Interesting real research
30
vi
DISCOVERING STATISTICS USING SAS
2
Everything you ever wanted to know about statistics  
(well, sort of)
31

2.1.
What will this chapter tell me? 1
31

2.2.
Building statistical models 1
32

2.3.
Populations and samples 1
34

2.4.
Simple statistical models 1
35


2.4.1. The mean: a very simple statistical model 1
35


2.4.2.  Assessing the fit of the mean: sums of squares, variance and standard  
deviations 1
35


2.4.3. Expressing the mean as a model 2
38

2.5.
Going beyond the data 1
40


2.5.1.
The standard error 1
40


2.5.2. Confidence intervals 2
43

2.6.
Using statistical models to test research questions 1
48


2.6.1. Test statistics 1
52


2.6.2. One- and two-tailed tests 1
54


2.6.3.
Type I and Type II errors 1
55


2.6.4. Effect sizes 2
56


2.6.5. Statistical power 2
58


What have I discovered about statistics? 1
59


Key terms that I’ve discovered
59


Smart Alex’s tasks
59


Further reading
60


Interesting real research
60
3
The SAS environment
61

3.1.
What will this chapter tell me? 1
61

3.2.
Versions of SAS 1
62

3.3.
Getting started 1
62

3.4.
Libraries 1     
63

3.5.
The program editor: getting data into SAS 1
65               


3.5.1.
Entering words and numbers into SAS with the  program editor 1
66


3.5.2.
Entering dates 1
69

3.6.
Entering data with Excel 1
71


3.6.1.
Saving the data 1
72


3.6.2.
Date formats 1
72


3.6.3.
Missing values 1
73

3.7.
The DATA step 1
73

3.8.
SAS formats 1
74


3.8.1.
Built in formats 1
74


3.8.2.
User-defined formats 1
76

3.9.
Variable labels 1
77

3.10. More on DATA steps 1
79


3.10.1. Calculating a new variable 1
79


3.10.2. Conditional (If ...then...) statements 1
79


3.10.3. Getting rid of variables 1
81


3.10.4. Getting rid of cases 1
81
vii
CONTENTS

3.11.
Procs for checking data 1
81

3.12.
Output and results 1
82

3.13.
Looking at data with PROC COnTEnTS 1
85

3.14. What have I discovered about statistics? 1
85

3.15.
Key terms that I’ve discovered  
86

3.16.
Smart Alex’s tasks  
86

3.17.
Further reading
87
4
Exploring data with graphs
88

4.1.
What will this chapter tell me? 1
88

4.2.
The art of presenting data 1
88


4.2.1. What makes a good graph? 1
89


4.2.2. Lies, damned lies, and … erm … graphs 1
91

4.3.
Charts in SAS 1
92

4.4.
Histograms: a good way to spot obvious problems 1
93

4.5.
Boxplots (box–whisker diagrams) 1
95

4.6.
Graphing means: bar charts and error bars 1
100


4.6.1. Simple bar charts for independent means 1
100


4.6.2. Clustered bar charts for independent means 1
101


4.6.3. Simple bar charts for related means 1
102


4.6.4. Clustered bar charts for ‘mixed’ designs 1
104

4.7.
Graphing relationships: the scatterplot 1
106


4.7.1. Simple scatterplot 1
106


4.7.2. Grouped scatterplot 1
108


4.7.3. Simple and grouped 3-D scatterplots 1
108


4.7.4. Matrix scatterplot 1
110


What have I discovered about statistics? 1
111


Key terms that I’ve discovered 1
112


Smart Alex’s tasks  
112


Further reading  
112


Interesting real research  
112
5
Exploring assumptions
113

5.1.
What will this chapter tell me? 1
113

5.2.
What are assumptions? 1
114

5.3.
Assumptions of parametric data 1
114

5.4.
The assumption of normality 1
115


5.4.1.  Oh no, it’s that pesky frequency distribution again: checking  
normality visually 1
116


5.4.2. Quantifying normality with numbers 1
117


5.4.3. Exploring groups of data 1
121

5.5.
Testing whether a distribution is normal 1
127


5.5.1. Doing the Kolmogorov–Smirnov test on SAS 1
127


5.5.2. Output from the explore procedure 1
128


5.5.3. Reporting the K–S test 1
130

5.6.
Testing for homogeneity of variance 1
130


5.6.1. Levene’s test 1
131

5.7.
Correcting problems in the data 2
133
viii
DISCOVERING STATISTICS USING SAS


5.7.1. Dealing with outliers 2
133


5.7.2. Dealing with non-normality and unequal variances 2
134


5.7.3. Transforming the data using SAS 2
136


5.7.4. When it all goes horribly wrong 3
140


What have I discovered about statistics? 1
142


Key terms that I’ve discovered
143


Smart Alex’s tasks
143


Further reading
143
6
Correlation
144

6.1.
What will this chapter tell me? 1
144

6.2.
Looking at relationships 1
145

6.3.
How do we measure relationships? 1
145


6.3.1. A detour into the murky world of covariance 1
145


6.3.2.
Standardization and the correlation coefficient 1
147


6.3.3.
The significance of the correlation coefficient 3
149


6.3.4. Confidence intervals for r 3
150


6.3.5. A word of warning about interpretation: causality 1
151

6.4.
Data entry for correlation analysis using SAS 1
152

6.5.
Bivariate correlation 1
152


6.5.1. General procedure for running correlations on SAS 1
152


6.5.2. Pearson’s correlation coefficient 1
154


6.5.3. Spearman’s correlation coefficient 1
157


6.5.4. Kendall’s tau (non-parametric) 1
159

6.6.
Partial correlation 2
160


6.6.1.
The theory behind part and partial correlation 2
160


6.6.2.
Partial correlation using SAS 2
162


6.6.3.
Semi-partial (or part) correlations 2
163

6.7.
Comparing correlations 3
164


6.7.1. Comparing independent rs 3
164


6.7.2. Comparing dependent rs 3
165

6.8.
Calculating the effect size 1
166

6.9.
How to report correlation coefficents 1
166


What have I discovered about statistics? 1
168


Key terms that I’ve discovered
169


Smart Alex’s tasks  
169


Further reading
170


Interesting real research
170
7
Regression
171

terryzhao1 发表于 2013-11-18 13:25
假的，别下载，请版主清理下！

7.1.
What will this chapter tell me? 1
171

7.2.
An introduction to regression 1
172


7.2.1. Some important information about straight lines 1
173


7.2.2.
The method of least squares 1
174


7.2.3.
Assessing the goodness of fit: sums of squares, R and R2 1
175


7.2.4. Assessing individual predictors 1
178
ix
CONTENTS

7.3.
Doing simple regression on SAS 1
179

7.4.
Interpreting a simple regression 1
180


7.4.1. Overall fit of the model 1
180


7.4.2. Model parameters 1
181


7.4.3. Using the model 1
182

7.5.
Multiple regression: the basics 2
183


7.5.1.
An example of a multiple regression model 2
184


7.5.2.
Sums of squares, R and R2 2
185


7.5.3. Methods of regression 2
186

7.6.
How accurate is my regression model? 2
188


7.6.1. Assessing the regression model I: diagnostics 2
188


7.6.2.
Assessing the regression model II: generalization 2
194

7.7.
How to do multiple regression using SAS 2
199


7.7.1.
Some things to think about before the analysis 2
199


7.7.2.
Main options 2
200


7.7.3.
Statistics 2
201


7.7.4. Saving regression diagnostics 2
202

7.8.
Interpreting multiple regression 2
203


7.8.1. Simple statistics 2
204


7.8.2.
Model parameters 2
207


7.8.3.
Comparing the models 1
211


7.8.4. Assessing the issue of multicollinearity 1
211


7.8.5. Casewise diagnostics 1
214


7.8.6.
Checking assumptions 2
216

7.9.
What if I violate an assumption? 2
221

7.10. How to report multiple regression 2
221

7.11. Categorical predictors and multiple regression 3
222


7.11.1. Dummy coding 3
222


7.11.2. SAS output for dummy variables 3
225


What have I discovered about statistics? 1
228


Key terms that I’ve discovered  
229


Smart Alex’s tasks  
229


Further reading  
230


Interesting real research  
230
8
Logistic regression
231

8.1.
What will this chapter tell me? 1
231

8.2.
Background to logistic regression 1
232

8.3.
What are the principles behind logistic regression? 3
232


8.3.1.
Assessing the model: the log-likelihood statistic 3
234


8.3.2.
Assessing the model: R, R2 and c 3
235


8.3.3.
Assessing the contribution of predictors: the Wald statistic 2
237


8.3.4. The odds ratio 3
238

8.4.
Assumptions and things that can go wrong 4
239


8.4.1. Assumptions 2
239


8.4.2. Incomplete information from the predictors 4
240


8.4.3. Complete separation 4
240

8.5.
Binary logistic regression: an example that will make you feel eel 2
242
x
DISCOVERING STATISTICS USING SAS


8.5.1.
The main analysis 2
243


8.5.2. Obtaining predicted probabilities and residuals 2
244


8.5.3. Final syntax
245

8.6.
Interpreting logistic regression 2
245


8.6.1. The initial output 2
245


8.6.2.
Intervention
246


8.6.3. Listing predicted probabilities 2
250


8.6.4. Interpreting residuals 2
251


8.6.5. Calculating the effect size 2
253

8.7.
How to report logistic regression 2
253

8.8.
Testing assumptions: another example 2
254


8.8.1. Testing for linearity of the logit 3
255


8.8.2. Testing the assumption of linearity of the logit
256

8.9.
Predicting several categories: multinomial logistic regression 3
258


8.9.1.
Running multinomial logistic regression in SAS 3
259


8.9.2. The final touches 3
260


8.9.3. Interpreting the multinomial logistic regression output 3
260


8.9.4. Reporting the results 3
264


What have I discovered about statistics? 1
265


Key terms that I’ve discovered
265


Smart Alex’s tasks
265


Further reading
267


Interesting real research
267
9
Comparing two means
268

9.1.
What will this chapter tell me? 1
268

9.2.
Looking at differences 1
269

9.3.
The t-test 1  
269


9.3.1. Two example data sets 1
270


9.3.2. Rationale for the t-test 1
271


9.3.3. Assumptions of the t-test 1
273

9.4.
The dependent t-test 1
273


9.4.1. Sampling distributions and the standard error 1
274


9.4.2. The dependent t-test equation explained 1
274


9.4.3. The dependent t-test and the assumption of normality 1
276


9.4.4. Dependent t-tests using SAS 1
276


9.4.5. Output from the dependent t-test 1
277


9.4.6. Calculating the effect size 2
278


9.4.7. Reporting the dependent t-test 1
279

9.5.
The independent t-test 1
280


9.5.1. The independent t-test equation explained 1
280


9.5.2. The independent t-test using SAS 1
284


9.5.3. Output from the independent t-test 1
285


9.5.4. Calculating the effect size 2
286


9.5.5. Reporting the independent t-test 1
287

9.6.
Between groups or repeated measures? 1
288

9.7.
The t-test as a general linear model 2
288

9.8.
What if my data are not normally distributed? 2
290


What have I discovered about statistics? 1
291


Key terms that I’ve discovered
291
xi
CONTENTS


Smart Alex’s task
291


Further reading
292


Interesting real research
292
10 Comparing several means: ANOVA (GLM 1)
293

10.1. What will this chapter tell me? 1
293

10.2. The theory behind AnOVA 2
294


10.2.1.
Inflated error rates 2
294


10.2.2.
Interpreting f 2
295


10.2.3.
AnOVA as regression 2
295


10.2.4.
Logic of the f-ratio 2
300


10.2.5.
Total sum of squares (SST) 2
302


10.2.6.
Model sum of squares (SSM) 2
302


10.2.7.
Residual sum of squares (SSR) 2
303


10.2.8.
Mean squares 2
304


10.2.9.
The f-ratio 2
304


10.2.10. Assumptions of AnOVA 3
305


10.2.11. Planned contrasts 2
306


10.2.12. Post hoc procedures 2
317

10.3. Running one-way AnOVA on SAS 2
319


10.3.1.
Planned comparisons using SAS 2
320


10.3.2.
Post hoc tests in SAS 2
322


10.3.3.
Options 2
322


10.3.4.
ODS graphics 2
323

10.4. Output from one-way AnOVA 2
324


10.4.1.
Output for the main analysis 2
324


10.4.2.
Output for trends and planned comparisons 2
327


10.4.3.
Output for post hoc tests 2
328


10.4.4.
Graph output for one-way AnOVA using PROC GLM 2
330

10.5. Calculating the effect size 2
331

10.6. Reporting results from one-way independent AnOVA 2
333

10.7. Violations of assumptions in one-way independent AnOVA 2
333


What have I discovered about statistics? 1
334


Key terms that I’ve discovered
334


Smart Alex’s tasks
335


Further reading
336


Interesting real research
336
11 Analysis of covariance, ANCOVA (GLM 2)
337

11.1. What will this chapter tell me? 2
337

11.2. What is AnCOVA? 2
338

11.3. Assumptions and issues in AnCOVA 3
339


11.3.1.
Independence of the covariate and treatment effect 3
339


11.3.2.
Homogeneity of regression slopes 3
341

11.4. Conducting AnCOVA on SAS 2
341


11.4.1.
Inputting data 1
341


11.4.2.  Initial considerations: testing the independence of the independent  
variable and covariate 2
343


11.4.3.
The main analysis 2
344
xii
DISCOVERING STATISTICS USING SAS


11.4.4.
Contrasts and other options 2
344

11.5. Interpreting the output from AnCOVA 2
345


11.5.1.
What happens when the covariate is excluded? 2
345


11.5.2.
The main analysis 2
346


11.5.3.
Contrasts 2
348


11.5.4.
Interpreting the covariate 2
349

11.6. AnCOVA run as a multiple regression 2
351

11.7. Testing the assumption of homogeneity of regression slopes 3
351

11.8. Calculating the effect size 2
353

11.9.
Reporting results 2
355

11.10. What to do when assumptions are violated in AnCOVA 3
356


What have I discovered about statistics? 2
356


Key terms that I’ve discovered
357


Smart Alex’s tasks
357


Further reading
358


Interesting real research
358
12 Factorial ANOVA (GLM 3)
359

12.1. What will this chapter tell me? 2
359

12.2. Theory of factorial AnOVA (between groups) 2
360


12.2.1.
Factorial designs 2
360


12.2.2.
An example with two independent variables 2
361


12.2.3.
Total sums of squares (SST) 2
362


12.2.4.
The model sum of squares (SSM) 2
364


12.2.5.
The residual sum of squares (SSR) 2
366


12.2.6.
The f-ratios 2
367

12.3. Factorial AnOVA using SAS 2
368


12.3.1.
Exploring the data: PROC MEAnS 2
368


12.3.2.
Contrasts and estimates 2
369


12.3.3.
Post hoc tests 2
370


12.3.4.
Simple effects 2
371


12.3.5.
ODS graphics 2
371

12.4. Output from factorial AnOVA 2
372


12.4.1.
The main AnOVA tables 2
372


12.4.2.
Contrasts  
375


12.4.3.
Least squares means and post hoc analysis 2
375


12.4.4.
Simple effects  
377


12.4.5.
Summary  
378

12.5. Interpreting interaction graphs 2
379

12.6. Calculating effect sizes 3
382

12.7. Reporting the results of two-way AnOVA 2
384

12.8. Factorial AnOVA as regression 3
384

12.9. What to do when assumptions are violated in factorial AnOVA 3
388


What have I discovered about statistics? 2
390


Key terms that I’ve discovered
390


Smart Alex’s tasks
390


Further reading
391


Interesting real research
392
xiii
CONTENTS
13 Repeated-measures designs (GLM 4)
393

13.1. What will this chapter tell me? 2
393

13.2. Introduction to repeated measures designs 2
394


13.2.1.
The assumption of sphericity 2
395


13.2.2.
How is sphericity measured? 2
395


13.2.3.
Assessing the severity of departures from sphericity 2
396


13.2.4.
What is the effect of violating the assumption of sphericity? 3
396


13.2.5.
What do you do if you violate sphericity? 2
397

13.3. Theory of one-way repeated measures AnOVA 2
398


13.3.1.
The total sum of squares (SST) 2
400


13.3.2.
The within-participant sum of squares (SSW) 2
401


13.3.3.
The model sum of squares (SSM) 2
402


13.3.4.
The residual sum of squares (SSR) 2
403


13.3.5.
The mean squares 2
403


13.3.6.
The f-ratio 2
403


13.3.7.
The between-participant sum of squares 2
404

13.4. One-way repeated-measures AnOVA using SAS 2
404


13.4.1.
The main analysis 2
404


13.4.2.
Defining contrasts for repeated measures 2
405

13.5. Output for one-way repeated-measures AnOVA 2
407


13.5.1.
Model description  
407


13.5.2.
Assessing and correcting for sphericity: Mauchly’s test 2
408


13.5.3.
The main AnOVA 2
409


13.5.4.
Contrasts 2
411

13.6. Effect sizes for repeated-measures AnOVA 3
414

13.7. Reporting one-way repeated-measures AnOVA 2
415

13.8. Repeated-measures with several independent variables 2
416


13.8.1.
Calculating and comparing means 2
418


13.8.2.
The main analysis 2
419

13.9. Output for factorial repeated-measures AnOVA 2
421


13.9.1.
Descriptive statistics 2
421


13.9.2.
Main analysis 2
422


13.9.3.
The effect of drink 2
424


13.9.4.
The effect of imagery 2
424


13.9.5.
The interaction effect (drink × imagery) 2
424


13.9.6.
Contrasts for repeated-measures variables 2
426

13.10. Effect sizes for factorial repeated-measures AnOVA 3
430

13.11. Reporting the results from factorial repeated-measures AnOVA 2
431

13.12. What to do when assumptions are violated in repeated-measures AnOVA 3 432


What have I discovered about statistics? 2
432


Key terms that I’ve discovered
433


Smart Alex’s tasks
433


Further reading  
434


Interesting real research
435

14 Mixed design ANOVA (GLM 5)
436

14.1. What will this chapter tell me? 1
436

14.2. Mixed designs 2
437
xiv
DISCOVERING STATISTICS USING SAS

14.3. What do men and women look for in a partner? 2
438

14.4. Mixed AnOVA on SAS 2
438

14.5. Output for mixed factorial AnOVA: main analysis 3
441


14.5.1.
The main effect of gender 2
443


14.5.2.
The main effect of looks 2
445


14.5.3.
The main effect of charisma 2
447


14.5.4.
The interaction between gender and looks 2
447


14.5.5.
The interaction between gender and charisma 2
449


14.5.6.
The interaction between attractiveness and charisma 2
550


14.5.7.
The interaction between looks, charisma and gender 3
453


14.5.8.
Conclusions 3
457

14.6. Calculating effect sizes 3
458

14.7. Reporting the results of mixed AnOVA 2
459

14.8. What to do when assumptions are violated in mixed AnOVA 3
462


What have I discovered about statistics? 2
462


Key terms that I’ve discovered
463


Smart Alex’s tasks
463


Further reading  
464


Interesting real research
464
15 Non-parametric tests
465

15.1. What will this chapter tell me? 1
465

15.2. When to use non-parametric tests 1
466

15.3.  Comparing two independent conditions: the Wilcoxon rank-sum test and  
Mann–Whitney test 1
466


15.3.1.
Theory 2  
468


15.3.2.
Inputting data and provisional analysis 1
471


15.3.3.
Running the analysis 1
473


15.3.4.
Output from the Wilcoxon two-sample test 1
474


15.3.5.
Calculating an effect size 2
476


15.3.6.
Writing the results 1
476

15.4. Comparing two related conditions: the Wilcoxon signed-rank test 1
478


15.4.1.
Theory of the Wilcoxon signed-rank test 2
478


15.4.2.
Running the analysis 1
481


15.4.3.
Output for the alcohol group 1
481


15.4.4.
Output for the ecstasy group 1
482


15.4.5.
Writing the results 1
482

15.5.
Differences between several independent groups: the Kruskal–Wallis test 1
483


15.5.1.
Theory of the Kruskal–Wallis test 2
484


15.5.2.
Inputting data and provisional analysis 1
486


15.5.3.
Doing the Kruskal–Wallis test on SAS 1
487


15.5.4.
Output from the Kruskal–Wallis test 1
487


15.5.5.
Post hoc tests for the Kruskal–Wallis test 2
488


15.5.6.
Writing and interpreting the results 1
492

15.6. Differences between several related groups: Friedman’s AnOVA 1
493


15.6.1.
Theory of Friedman’s AnOVA 2
494


15.6.2.
Inputting data and provisional analysis 1
495


15.6.3.
Doing Friedman’s AnOVA on SAS 1
496
xv
CONTENTS


15.6.4.
Output from Friedman’s AnOVA 1
497


15.6.5.
Post hoc tests for Friedman’s AnOVA 2
498


15.6.6.
Writing and interpreting the results 1
500


What have I discovered about statistics? 1
500


Key terms that I’ve discovered
501


Smart Alex’s tasks
501


Further reading  
502


Interesting real research
502
16 Multivariate analysis of variance (MANOVA)
503

16.1. What will this chapter tell me? 2
503

16.2. When to use MAnOVA 2
504

16.3. Introduction: similarities and differences to AnOVA 2
504


16.3.1.
Words of warning 2
506


16.3.2.
The example for this chapter 2
506

16.4. Theory of MAnOVA 3
507


16.4.1.
Introduction to matrices 3
507


16.4.2.
Some important matrices and their functions 3
509


16.4.3.
Calculating MAnOVA by hand: a worked example 3
510


16.4.4.
Principle of the MAnOVA test statistic 4
517

16.5. Practical issues when conducting MAnOVA 3
522


16.5.1.
Assumptions and how to check them 3
522


16.5.2.
Choosing a test statistic 3
523


16.5.3.
Follow-up analysis 3
524

16.6. MAnOVA on SAS 2
524


16.6.1.
The main analysis 2
525


16.6.2.
Multiple comparisons in MAnOVA 2
525

16.7. Output from MAnOVA 3
526


16.7.1.
Preliminary analysis and testing assumptions 3
526


16.7.2.
MAnOVA test statistics 3
527


16.7.3.
Univariate test statistics 2
528


16.7.4.
SSCP matrices 3
529


16.7.5.
Contrasts 3
530

16.8. Box’s test 3

531

16.9. Reporting results from MAnOVA 2
532

16.10. Following up MAnOVA with discriminant function analysis 3
533

16.11. Output from the discriminant function analysis 4
533

16.12. Reporting results from discriminant function analysis 2
536

16.13. Some final remarks 4
536


16.13.1. The final interpretation 4
536


16.13.2. Univariate AnOVA or discriminant function analysis?
537

16.14. What to do when assumptions are violated in MAnOVA 3
537


What have I discovered about statistics? 2
539


Key terms that I’ve discovered
539


Smart Alex’s tasks
540


Further reading
540


Interesting real research
540
xvi
DISCOVERING STATISTICS USING SAS
17 Exploratory factor analysis
541

17.1. What will this chapter tell me? 1
541

17.2. When to use factor analysis 2
542

17.3. Factors 2

542


17.3.1.
Graphical representation of factors 2
544


17.3.2.
Mathematical representation of factors 2
545


17.3.3.
Factor scores 2
547

17.4. Discovering factors 2
548


17.4.1.
Choosing a method 2
548


17.4.2.
Communality 2
549


17.4.3.
Factor analysis vs. principal component analysis 2
550


17.4.4.
Theory behind principal component analysis 3
551


17.4.5.
Factor extraction: eigenvalues and the scree plot 2
552


17.4.6.
Improving interpretation: factor rotation 3
554

17.5. Research example 2
557


17.5.1.
Before you begin 2
559

17.6. Running the analysis 2
562


17.6.1.
Factor extraction on SAS 2
562


17.6.2.
Rotation 2
563


17.6.3.
Saving factor scores  
563

17.7. Interpreting output from SAS  
564


17.7.1.
Preliminary analysis 2
566


17.7.2.
Factor extraction 2
567


17.7.3.
Factor rotation 2
573


17.7.4.
Factor scores 2
578


17.7.5.
Summary 2
579

17.8. How to report factor analysis 1
579

17.9. Reliability analysis 2
582


17.9.1.
Measures of reliability 3
582


17.9.2.
Interpreting Cronbach’s α (some cautionary tales) 2
583


17.9.3.
Reliability analysis on SAS 2
585


17.9.4.
Interpreting the output 2
585

17.10. How to report reliability analysis 2
589


What have I discovered about statistics? 2
590


Key terms that I’ve discovered
590


Smart Alex’s tasks
591


Further reading  
593


Interesting real research
593
18 Categorical data
594

18.1. What will this chapter tell me? 1
594

18.2. Analysing categorical data 1
595

18.3. Theory of analysing categorical data 1
595


18.3.1.
Pearson’s chi-square test 1
596


18.3.2.
Fisher’s exact test 1
598


18.3.3.
The likelihood ratio 2
598


18.3.4.
Yates’s correction 2
599
xvii
CONTENTS

18.4. Assumptions of the chi-square test 1
599

18.5. Doing chi-square on SAS 1
600


18.5.1.
Entering data: raw scores 1
600


18.5.2.
Entering data: weight cases 1
600


18.5.3.
Running the analysis 1
601


18.5.4.
Output for the chi-square test 1
601


18.5.5.  Breaking down a significant chi-square test with standardized residuals 2
604


18.5.6.
Calculating an effect size 2
606


18.5.7.
Reporting the results of chi-square 1
607

18.6. Several categorical variables: loglinear analysis 3
609


18.6.1.
Chi-square as regression 4
609


18.6.2.
Loglinear analysis 3
615

18.7. Assumptions in loglinear analysis 2
617

18.8. Loglinear analysis using SAS 2
617


18.8.1.
Initial considerations 2
617


18.8.2.
The loglinear analysis 2
619

18.9. Output from loglinear analysis 3
620

18.10. Following up loglinear analysis 2
624

18.11. Effect sizes in loglinear analysis 2
625

18.12. Reporting the results of loglinear analysis 2
626


What have I discovered about statistics? 1
626


Key terms that I’ve discovered
627


Smart Alex’s tasks  
627


Further reading  
628


Interesting real research
628
19 Multilevel linear models
629

19.1. What will this chapter tell me? 1
629

19.2. Hierarchical data 2
630


19.2.1.
The intraclass correlation 2
632


19.2.2.
Benefits of multilevel models 2
633

19.3. Theory of multilevel linear models 3
634


19.3.1.
An example 2
634


19.3.2.
Fixed and random coefficients 3
636

19.4. The multilevel model 4
638


19.4.1.
Assessing the fit and comparing multilevel models 4
641


19.4.2.
Types of covariance structures 4
641

19.5. Some practical issues 3
643


19.5.1.
Assumptions 3
643


19.5.2.
Sample size and power 3
644


19.5.3.
Centring variables 4
644

19.6. Multilevel modelling on SAS 4
645


19.6.1.
Entering the data 2
646


19.6.2.
Ignoring the data structure: AnOVA 2
646


19.6.3.
Ignoring the data structure: AnCOVA 2
648


19.6.4.
Factoring in the data structure: random intercepts 3
650


19.6.5.
Factoring in the data structure: random intercepts and slopes 4
652


19.6.6.
Adding an interaction to the model 4
659
xviii
DISCOVERING STATISTICS USING SAS

19.7. Growth models 4
662


19.7.1.
Growth curves (polynomials) 4
662


19.7.2.
An example: the honeymoon period 2
663


19.7.3.
Restructuring the data 3
664


19.7.4.
Running a growth model on SAS 4
667


19.7.5.
Further analysis 4
670

19.8. How to report a multilevel model 3
672


What have I discovered about statistics? 2
673


Key terms that I’ve discovered
674


Smart Alex’s tasks
674


Further reading  
674


Interesting real research
675
Epilogue: Life after Discovering Statistics
676
Glossary
679
Appendix
695

A.1.
Table of the standard normal distribution
695

A.2.
Critical values of the t-distribution
701

A.3.
Critical values of the f-distribution
702

A.4.
Critical values of the chi-square distribution
706
References
707
Index
713

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