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Table of Contents
Introduction................................... 1
1 Measuring Variation: How Values Differ.......................... 5
How Variation Is Measured...........................................5
Sum of Deviations..........................................................6
Summing Squared Deviations...............................................7
From the Sum of Squares to the Variance................................10
Using the VAR.P( ) and VAR.S( ) Functions....................................11
The Standard Deviation................................................14
The Standard Error of the Mean............................................15
About z-Scores and z-Values.................................................18
About t-Values.....................................................................23
2 Correlation.........................................29
Measuring Correlation...........................................................................29
Expressing the Strength of a Correlation.....................30
Determining a Correlation’s Direction...................................32
Calculating Correlation.......................................................34
Step One: The Covariance..................................34
Watching for Signs........................................................36
From the Covariance to the Correlation Coefficient..........................38
Using the CORREL( ) Function...................................................41
Understanding Bias in the Correlation............................41
Checking for Linearity and Outliers in the Correlation ........................44
Avoiding a Trap in Charting.............................48
Correlation and Causation..............................................53
Direction of Cause........................................54
A Third Variable................................................55
Restriction of Range..........................................................................55
3 Simple Regression.....................................59
Predicting with Correlation and Standard Scores.........................60
Calculating the Predictions............................61
Returning to the Original Metric............................63
Generalizing the Predictions........................................64
Predicting with Regression Coefficient and Intercept.................................65
The SLOPE( ) Function........................................................65
The INTERCEPT( ) Function.....................69
Charting the Predictions....................................70
Shared Variance...........................................71
The Standard Deviation, Reviewed.............................71
More About Sums of Squares..................................73
Sums of Squares Are Additive..............................................74
R2 in Simple Linear Regression.........................................77
Sum of Squares Residual versus Sum of Squares Within.......................81
The TREND( ) Function............................................82
Array-entering TREND( )..........................................84
TREND( )’s new x’s Argument..................................85
TREND( )’s const Argument...................................................86
Calculating the Zero-constant Regression.............................88
Partial and Semipartial Correlations..........................90
Partial Correlation............................................91
Understanding Semipartial Correlations........................................................95
4 Using the LINEST( ) Function...........................103
Array-Entering LINEST( ).............................. 103
Understanding the Mechanics of Array Formulas.....................104
Inventorying the Mistakes............................................105
Comparing LINEST( ) to SLOPE( ) and INTERCEPT( )..........................107
The Standard Error of a Regression Coefficient..................................109
The Meaning of the Standard Error of a Regression Coefficient........................109
A Regression Coefficient of Zero......................................................110
Measuring the Probability That the Coefficient is Zero in the Population...............112
Statistical Inference as a Subjective Decision............................113
The t-ratio and the F-ratio..............................116
Interval Scales and Nominal Scales.............................116
The Squared Correlation, R2.....................................117
The Standard Error of Estimate...........................120
The t Distribution and Standard Errors.......................121
Standard Error as a Standard Deviation of Residuals..............125
Homoscedasticity: Equal Spread................................128
Understanding LINEST( )’s F-ratio....................129
he Analysis of Variance and the F-ratio in Traditional Usage......................129
The Analysis of Variance and the F-ratio in Regression.........................131
Partitioning the Sums of Squares in Regression.....................133
The F-ratio in the Analysis of Variance........................................136
The F-ratio in Regression Analysis..................................................140
The F-ratio Compared to R2............................................................................146
The General Linear Model, ANOVA, and Regression Analysis........................146
Other Ancillary Statistics from LINEST( ).....................................149
5 Multiple Regression...................................151
A Composite Predictor Variable.........................152
Generalizing from the Single to the Multiple Predictor........................153
Minimizing the Sum of the Squared Errors.......................................156
Understanding the Trendline...........................................................160
Mapping LINEST( )’s Results to the Worksheet......................................163
Building a Multiple Regression Analysis from the Ground Up......................166
Holding Variables Constant............................................166
Semipartial Correlation in a Two-Predictor Regression................167
Finding the Sums of Squares....................................169
R2 and Standard Error of Estimate......................................170
F-Ratio and Residual Degrees of Freedom.................................172
Calculating the Standard Errors of the Regression Coefficients...........................173
Some Further Examples................................................176
Using the Standard Error of the Regression Coefficient..........................181
Arranging a Two-Tailed Test....................................186
Arranging a One-Tailed Test.....................................189
Using the Models Comparison Approach to Evaluating Predictors...................192
Obtaining the Models’ Statistics.......................................192
Using Sums of Squares Instead of R2............................196
Estimating Shrinkage in R2..................................................197
6 Assumptions and Cautions Regarding Regression Analysis................199
About Assumptions.................................................199
Robustness: It Might Not Matter...................................202
Assumptions and Statistical Inference.................................204
The Straw Man............................................................................204
Coping with Nonlinear and Other Problem Distributions.........................211
The Assumption of Equal Spread...........................................213
Using Dummy Coding..........................................215
Comparing the Regression Approach to the t-test Approach..................217
Two Routes to the Same Destination.....................................218
Unequal Variances and Sample Sizes..................................220
Unequal Spread: Conservative Tests..........................................220
Unequal Spread: Liberal Tests.............................................................225
Unequal Spreads and Equal Sample Sizes.........................226
Using LINEST()Instead of the Data Analysis Tool......................................230
Understanding the Differences Between the T.DIST()Functions........................231
Using Welch’s Correction................................237
The TTEST()Function................................................243
7 Using Regression to Test Differences Between Group Means.........................245
Dummy Coding.............................................................246
An Example with Dummy Coding....................................246
Populating the Vectors Automatically.....................................250
The Dunnett Multiple Comparison Procedure..........................253
Effect Coding...................................................................259
Coding with -1 Instead of 0.........................................260
Relationship to the General Linear Model..............................261
Multiple Comparisons with Effect Coding...............................264
Orthogonal Coding................................................267
Establishing the Contrasts................................267
Planned Orthogonal Contrasts Via ANOVA..........................268
Planned Orthogonal Contrasts Using LINEST( )...........................269
Factorial Analysis.......................................................272
Factorial Analysis with Orthogonal Coding....................274
Factorial Analysis with Effect Coding..............................279
Statistical Power, Type I and Type II Errors.....................283
Calculating Statistical Power..............................285
Increasing Statistical Power...........................................286
Coping with Unequal Cell Sizes.......................................288
Using the Regression Approach...............................289
Sequential Variance Assignment...............................................291
8 The Analysis of Covariance..............................295
Contrasting the Results.............................................297
ANCOVA Charted................................305
Structuring a Conventional ANCOVA......................308
Analysis Without the Covariate....................308
Analysis with the Covariate..............................310
Structuring an ANCOVA Using Regression.......................315
Checking for a Common Regression Line..........................316
Summarizing the Analysis...............................320
Testing the Adjusted Means: Planned Orthogonal Coding in ANCOVA...............321
ANCOVA and Multiple Comparisons Using the Regression Approach.......................328
Multiple Comparisons via Planned Nonorthogonal Contrasts..................................330
Multiple Comparisons with Post Hoc Nonorthogonal Contrasts...............................332
TOC, 9780789756558, 4/13/2016
