Repeated Data Contrasts • EASI

This page examines a single-factor within-subjects (repeated measures) design using raw data input, focusing on comparisons and contrasts.

Preliminary Tasks

Data Entry

This code inputs the variable names and creates a viewable data frame.

Outcome1 <- c(6, 8, 6, 8, 10, 8, 10, 9, 8, 7)
Outcome2 <- c(7, 13, 11, 10, 13, 8, 11, 14, 12, 11)
Outcome3 <- c(9, 16, 11, 12, 15, 13, 9, 14, 11, 10)
RepeatedData <- construct(Outcome1, Outcome2, Outcome3)

Summary Statistics

This code obtains the descriptive statistics for the data frame.

(RepeatedData) |> describeMoments()


Summary Statistics for the Data

               N       M      SD    Skew    Kurt
Outcome1  10.000   8.000   1.414   0.000  -0.738
Outcome2  10.000  11.000   2.211  -0.617  -0.212
Outcome3  10.000  12.000   2.449   0.340  -1.102

This code will display the correlations among the variables.

(RepeatedData) |> describeCorrelations()


Correlations for the Data

         Outcome1 Outcome2 Outcome3
Outcome1   1.000    0.533    0.385 
Outcome2   0.533    1.000    0.574 
Outcome3   0.385    0.574    1.000

Analyses of the Means

This section produces analyses that are equivalent to one-sample analyses separately for each level of a factor.

Confidence Intervals

This code will provide a table of confidence intervals for each level of the factor.

(RepeatedData) |> estimateMeans()


Confidence Intervals for the Means

             Est      SE      df      LL      UL
Outcome1   8.000   0.447   9.000   6.988   9.012
Outcome2  11.000   0.699   9.000   9.418  12.582
Outcome3  12.000   0.775   9.000  10.248  13.752

This code will produce a graph of the confidence intervals for each level of the factor.

(RepeatedData) |> plotMeans()

The code defaults to 95% confidence intervals. This can be changed if desired.

(RepeatedData) |> estimateMeans(conf.level = .99)


Confidence Intervals for the Means

             Est      SE      df      LL      UL
Outcome1   8.000   0.447   9.000   6.547   9.453
Outcome2  11.000   0.699   9.000   8.728  13.272
Outcome3  12.000   0.775   9.000   9.483  14.517

For the graph, it is possible to add a comparison line to represent a population (or test) value and a region of practical equivalence in addition to changing the confidence level.

(RepeatedData) |> plotMeans(conf.level = .99, line = 9, rope = c(8, 10))

Significance Tests

This code will produce a table of NHST separately for each level of the factor. In this case, all the means are tested against a value of zero.

(RepeatedData) |> testMeans()


Hypothesis Tests for the Means

            Diff      SE      df       t       p
Outcome1   8.000   0.447   9.000  17.889   0.000
Outcome2  11.000   0.699   9.000  15.732   0.000
Outcome3  12.000   0.775   9.000  15.492   0.000

Often, the default test value of zero is not meaningful or plausible. This too can be altered (often in conjunction with what is presented in the plot).

(RepeatedData) |> testMeans(mu = 9)


Hypothesis Tests for the Means

            Diff      SE      df       t       p
Outcome1  -1.000   0.447   9.000  -2.236   0.052
Outcome2   2.000   0.699   9.000   2.860   0.019
Outcome3   3.000   0.775   9.000   3.873   0.004

Standardized Effect Sizes

This code will produce a table of standardized mean differences separately for each level of the factor. In this case, the mean is compared to zero to form the effect size.

(RepeatedData) |> standardizeMeans()


Confidence Intervals for the Standardized Means

               d      SE      LL      UL
Outcome1   5.657   1.251   3.005   8.295
Outcome2   4.975   1.111   2.622   7.312
Outcome3   4.899   1.096   2.579   7.203

Here too it is possible to alter the width of the confidence intervals and to establish a more plausible comparison value for the mean.

(RepeatedData) |> standardizeMeans(mu = 9, conf.level = .99)


Confidence Intervals for the Standardized Means

               d      SE      LL      UL
Outcome1  -0.707   0.364  -1.614   0.222
Outcome2   0.905   0.384  -0.083   1.873
Outcome3   1.225   0.422   0.126   2.317

Analyses of a Comparison

This section produces analyses involving comparisons of two levels of a factor.

Confidence Intervals

This code identifies the two levels for comparison and estimates the confidence interval of the difference.

(RepeatedData) |> focus(Outcome1, Outcome2) |> estimateDifference()


Confidence Interval for the Mean Difference

               Est      SE      df      LL      UL
Comparison   3.000   0.596   9.000   1.651   4.349

This code obtains and plots the confidence intervals for the mean difference in the identified comparison.

(RepeatedData) |> focus(Outcome1, Outcome2) |> plotDifference()

Of course, you can change the confidence level from the default 95% if desired.

(RepeatedData) |> focus(Outcome1, Outcome2) |> estimateDifference(conf.level = .99)


Confidence Interval for the Mean Difference

               Est      SE      df      LL      UL
Comparison   3.000   0.596   9.000   1.062   4.938

Once again, the confidence levels can be changed away from the default and a comparison line to represent a population (or test) value and a region of practical equivalence can be added to the graph..

(RepeatedData) |> focus(Outcome1, Outcome2) |> plotDifference(conf.level = .99, line = 0, rope = c(-2, 2))

If you wish, you can get the confidence intervals for the means and the mean difference in one command.

(RepeatedData) |> focus(Outcome1, Outcome2) |> estimateComparison()


Confidence Intervals for the Mean Comparison

               Est      SE      df      LL      UL
Outcome1     8.000   0.447   9.000   6.988   9.012
Outcome2    11.000   0.699   9.000   9.418  12.582
Comparison   3.000   0.596   9.000   1.651   4.349

This code produces a difference plot using the confidence intervals for the means and the mean difference.

(RepeatedData) |> focus(Outcome1, Outcome2) |> plotComparison()

Of course, you can change the confidence level from the default 95% if desired.

(RepeatedData) |> focus(Outcome1, Outcome2) |> estimateComparison(conf.level = .99)


Confidence Intervals for the Mean Comparison

               Est      SE      df      LL      UL
Outcome1     8.000   0.447   9.000   6.547   9.453
Outcome2    11.000   0.699   9.000   8.728  13.272
Comparison   3.000   0.596   9.000   1.062   4.938

Once again, the confidence levels can be changed away from the default and a region of practical equivalence can be added to the graph.

(RepeatedData) |> focus(Outcome1, Outcome2) |> plotComparison(conf.level = .99, rope = c(-2, 2))

Significance Test

This code produces NHST for the identified comparison (using a default test value of zero).

(RepeatedData) |> focus(Outcome1, Outcome2) |> testDifference()


Hypothesis Test for the Mean Difference

              Diff      SE      df       t       p
Comparison   3.000   0.596   9.000   5.031   0.001

If the default value of zero is not plausible, it too can be changed.

(RepeatedData) |> focus(Outcome1, Outcome2) |> testDifference(mu = -2)


Hypothesis Test for the Mean Difference

              Diff      SE      df       t       p
Comparison   5.000   0.596   9.000   8.385   0.000

Standardized Effect Size

This code calculates a standardized mean difference for the comparison and its confidence interval.

(RepeatedData) |> focus(Outcome1, Outcome2) |> standardizeDifference()


Confidence Interval for the Standardized Mean Difference

                 d      SE      LL      UL
Comparison   1.616   0.466   0.703   2.530

The width of the confidence interval for the effect size can be altered if desired.

(RepeatedData) |> focus(Outcome1, Outcome2) |> standardizeDifference(conf.level = .99)


Confidence Interval for the Standardized Mean Difference

                 d      SE      LL      UL
Comparison   1.616   0.466   0.416   2.816

Analyses of a Contrast

This section produces analyses involving multiple levels of a factor.

Confidence Intervals

This code produces a confidence interval for a specified contrast.

(RepeatedData) |> estimateContrast(contrast = c(-1, .5, .5))


Confidence Interval for the Mean Contrast

             Est      SE      df      LL      UL
Contrast   3.500   0.573   9.000   2.205   4.795

This code obtains and plots the confidence intervals for the mean difference in the identified contrast.

(RepeatedData) |> plotContrast(contrast = c(-1, .5, .5))

As in all other cases, the default value of the confidence interval can be changed.

(RepeatedData) |> estimateContrast(contrast = c(-1, .5, .5), conf.level = .99)


Confidence Interval for the Mean Contrast

             Est      SE      df      LL      UL
Contrast   3.500   0.573   9.000   1.639   5.361

The width of the confidence interval for the contrast can be altered and a comparison line to represent a population (or test) value and a region of practical equivalence can be added to the graph.

(RepeatedData) |> plotContrast(contrast = c(-1, .5, .5), conf.level = .99, line = 0, rope = c(-2, 2))

If you wish, you can get the confidence intervals for the mean subsets and the mean contrast in one command.

(RepeatedData) |> estimateSubsets(contrast = c(-1, .5, .5))


Confidence Intervals for the Mean Subsets

                 Est      SE      df      LL      UL
Neg Weighted   8.000   0.447   9.000   6.988   9.012
Pos Weighted  11.500   0.654   9.000  10.020  12.980
Contrast       3.500   0.573   9.000   2.205   4.795

This code produces a difference plot using the confidence intervals for the mean subsets and the mean contrast.

(RepeatedData) |> plotSubsets(contrast = c(-1, .5, .5))

Of course, you can change the confidence level from the default 95% if desired.

(RepeatedData) |> estimateSubsets(contrast = c(-1, .5, .5), conf.level = .99)


Confidence Intervals for the Mean Subsets

                 Est      SE      df      LL      UL
Neg Weighted   8.000   0.447   9.000   6.547   9.453
Pos Weighted  11.500   0.654   9.000   9.374  13.626
Contrast       3.500   0.573   9.000   1.639   5.361

Once again, the confidence levels can be changed away from the default and a region of practical equivalence can be added to the graph.

(RepeatedData) |> plotSubsets(contrast = c(-1, .5, .5), labels = c("Outcome1", "Others"), conf.level = .99, rope = c(-2, 2))

Significance Test

This code produces a NHST for the identified contrast. It tests the contrast against a value of zero by default.

(RepeatedData) |> testContrast(contrast = c(-1, .5, .5))


Hypothesis Test for the Mean Contrast

             Est      SE      df       t       p
Contrast   3.500   0.573   9.000   6.113   0.000

If desired, the contrast can be tested against other values.

(RepeatedData) |> testContrast(contrast = c(-1, .5, .5), mu = 4)


Hypothesis Test for the Mean Contrast

             Est      SE      df       t       p
Contrast  -0.500   0.573   9.000  -0.873   0.405

Standardized Effect Size

This code calculates a standardized contrast and its confidence interval.

(RepeatedData) |> standardizeContrast(contrast = c(-1, .5, .5))


Confidence Interval for the Standardized Mean Contrast

             Est      SE      LL      UL
Contrast   1.689   0.371   0.962   2.415

The width of the confidence interval for the effect size can be altered if desired.

(RepeatedData) |> standardizeContrast(contrast = c(-1, .5, .5), conf.level = .99)


Confidence Interval for the Standardized Mean Contrast

             Est      SE      LL      UL
Contrast   1.689   0.371   0.734   2.643

Analyses of a Complex Contrast

This section examines a complex contrast among multiple means.

Confidence Intervals

Create a single contrast to compare the first variable to the grand mean (which requires some arithmetic). Then esimate and plot the contrast.

(RepeatedData) |> estimateContrast(contrast = c(2/3, -1/3, -1/3))


Confidence Interval for the Mean Contrast

             Est      SE      df      LL      UL
Contrast  -2.333   0.382   9.000  -3.197  -1.470

(RepeatedData) |> plotContrast(contrast = c(2/3, -1/3, -1/3))

Rather than setting just one contrast, set two contrasts: one for the Grand Mean and one for Level 1. Estimate and plot the confidence intervals for each contrast and the difference between contrasts.

(RepeatedData) |> estimateComplex(contrast1 = c(1/3, 1/3, 1/3), contrast2 = c(1, 0, 0), labels = c("GrandMean", "O1Only"))


Confidence Intervals for the Mean Contrasts

              Est      SE      df      LL      UL
GrandMean  10.333   0.528   9.000   9.138  11.528
O1Only      8.000   0.447   9.000   6.988   9.012
Contrast   -2.333   0.382   9.000  -3.197  -1.470

(RepeatedData) |> plotComplex(contrast1 = c(1/3, 1/3, 1/3), contrast2 = c(1, 0, 0), labels = c("GrandMean", "O1Only"))

Significance Tests

The two versions of the contrast can be tested for statistical significance.

(RepeatedData) |> testContrast(contrast = c(2/3, -1/3, -1/3))


Hypothesis Test for the Mean Contrast

             Est      SE      df       t       p
Contrast  -2.333   0.382   9.000  -6.113   0.000

(RepeatedData) |> testComplex(contrast1 = c(1/3, 1/3, 1/3), contrast2 = c(1, 0, 0), labels = c("GrandMean", "L1Only"))


Hypothesis Tests for the Mean Contrasts

              Est      SE      df       t       p
GrandMean  10.333   0.528   9.000  19.563   0.000
L1Only      8.000   0.447   9.000  17.889   0.000
Contrast   -2.333   0.382   9.000  -6.113   0.000