Bivariate Summary Regression • EASI

This page analyzes bivariate regression models using summary statistics input.

Preliminary Tasks

Data Entry

This code inputs the variable summaries and creates a summary table. Note that analyses assume that the second variable is the criterion and the first variable the predictors.

Predictor <- c(N = 10, M = 8.000, SD = 1.414)
Criterion <- c(N = 10, M = 11.000, SD = 2.211)
BivariateSummary <- construct(Predictor, Criterion, class = "wsm")

This code creates a correlation matrix.

Predictor <- c(1.000, .533)
Criterion <- c(.533, 1.000)
BivariateCorr <- construct(Predictor, Criterion, class = "cor")

Summary Statistics

This code confirms the descriptive statistics from the summary tables.

(BivariateSummary) |> describeMoments()


Summary Statistics for the Data

                N       M      SD
Predictor  10.000   8.000   1.414
Criterion  10.000  11.000   2.211

(BivariateCorr) |> describeCorrelations()


Correlations for the Data

          Predictor Criterion
Predictor   1.000     0.533  
Criterion   0.533     1.000

Analyses of Overall Model

This section produces analyses of the overall regression model.

This code will produce a source table associated with the regression model.

(BivariateSummary) |> describeModel(BivariateCorr)


Source Table for the Regression Model

           SS      df      MS
Model  12.499   1.000  12.499
Error  31.498   8.000   3.937
Total  43.997   9.000   4.889

Confidence Interval

This code will produce the confidence interval for R Squared.

(BivariateSummary) |> estimateModel(BivariateCorr)


Proportion of Variance Accounted For by the Regression Model

          Est      LL      UL
Model   0.284   0.000   0.555

The code defaults to 90% confidence intervals. This can be changed if desired.

(BivariateSummary) |> estimateModel(BivariateCorr, conf.level = .95)


Proportion of Variance Accounted For by the Regression Model

          Est      LL      UL
Model   0.284   0.000   0.600

Significance Test

This code will calculate NHST for the regression model.

(BivariateSummary) |> testModel(BivariateCorr)


Hypothesis Test for the Regression Model

            F     df1     df2       p
Model   3.175   1.000   8.000   0.113

Analyses of the Regression Coefficients

This section analyses the regression coefficients obtained from the overall model.

Confidence Intervals

This code will provide a table of confidence intervals for each of the regression coefficients.

(BivariateSummary) |> estimateCoefficients(BivariateCorr)


Confidence Intervals for the Regression Coefficients

                Est      SE      LL      UL
(Intercept)   4.333   3.794  -4.417  13.082
Predictor     0.833   0.468  -0.245   1.912

This code will produce a graph of the confidence intervals for each of the regression coefficients.

(BivariateSummary) |> plotCoefficients(BivariateCorr)

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

(BivariateSummary) |> estimateCoefficients(BivariateCorr, conf.level = .99)


Confidence Intervals for the Regression Coefficients

                Est      SE      LL      UL
(Intercept)   4.333   3.794  -8.399  17.064
Predictor     0.833   0.468  -0.736   2.403

For the graph, it is possible to plot just coefficients for the predictors (minus the intercept) in addition to changing the confidence level. A comparison line and region of practical equivalence can also be added.

(BivariateSummary) |> plotCoefficients(BivariateCorr, conf.level = .99, line = 0, rope = c(-.5, .5), intercept = FALSE)

Significance Tests

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

(BivariateSummary) |> testCoefficients(BivariateCorr)


Hypothesis Tests for the Regression Coefficients

                Est      SE       t       p
(Intercept)   4.333   3.794   1.142   0.287
Predictor     0.833   0.468   1.782   0.113

Standardized Coefficient

This code will provide a table of confidence intervals for the standardized coefficient.

(BivariateSummary) |> standardizeCoefficients(BivariateCorr)


Confidence Intervals for the Standardized Regression Coefficients

              Est      SE      LL      UL
Predictor   0.833   0.468  -0.245   1.912

As in other places, the code defaults to a 95% confidence interval. This can be changed if desired.

(BivariateSummary) |> standardizeCoefficients(BivariateCorr, conf.level = .99)


Confidence Intervals for the Standardized Regression Coefficients

              Est      SE      LL      UL
Predictor   0.833   0.468  -0.736   2.403

Analyses of Regression Values

This section provides analyses of individual predicted values.

Regression Line

This code produces a plot of the regression line (with confidence and prediction intervals suppressed).

(BivariateSummary) |> plotLine(BivariateCorr, interval = "none")

Confidence and Prediction Intervals

This code provides estimates confidence and prediction limits for a specific value of the Predictor (value=4).

(BivariateSummary) |> estimateLine(BivariateCorr, value = 8)


Confidence and Prediction Intervals for the Regression

      Est   CI.LL   CI.UL   PI.LL   PI.UL
8  11.000   9.553  12.447   6.201  15.799

This code plots the confidence interval associated with the regression line and labels the interval for the specific value of the Predictor.

(BivariateSummary) |> plotLine(BivariateCorr, value = 8, interval = "confidence")

This code plots the prediction interval associated with the regression line and labels the interval for the specific value of the Predictor.

(BivariateSummary) |> plotLine(BivariateCorr, value = 8, interval = "prediction")