Correlations Multivariate This correlation matrix is calculated by the method that you select in the launch window. Correlation Probability Shows the Correlation Probability report, which is a matrix of p-values. Each p-value corresponds to a test of the null hypothesis that the true correlation between the variables is zero. This is a test of no linear relationship between the two response variables. The test is the usual test for significance of the Pearson correlation coefficient. CI of Correlation Shows the two-tailed confidence intervals of the correlations. This option is off by default. The default confidence coefficient is 95%. Use the Set α Level option to change the confidence coefficient. Inverse Correlations Shows or hides the inverse correlation matrix (Inverse Corr table). This option is off by default. The diagonal elements of the matrix are a function of how closely the variable is a linear function of the other variables. In the inverse correlation, the diagonal is 1/(1 – R2) for the fit of that variable by all the other variables. If the multiple correlation is zero, the diagonal inverse element is 1. If the multiple correlation is 1, then the inverse element becomes infinite and is reported missing. Partial Correlations Shows or hides the partial correlation table (Partial Corr), which shows the measure of the relationship between a pair of variables after adjusting for the effects of all the other variables. This option is off by default. This table is the negative of the inverse correlation matrix, scaled to unit diagonal. Covariance Matrix Shows or hides the covariance matrix which measures the degree to which a pair of variables change together. This option is off by default. Pairwise Correlations Shows or hides the Pairwise Correlations table, which lists the Pearson product-moment correlations for each pair of Y variables. This option is off by default. The correlations are calculated by the pairwise deletion method. The count values differ if any pair has a missing value for either variable. The Pairwise Correlations report also shows significance probabilities and compares the correlations in a bar chart. All results are based on the pairwise method. Hotelling’s T2 Test Allows you to conduct a one-sample test for the mean of the multivariate distribution of the variables that you entered as Y. Specify the mean vector under the null hypothesis in the window that appears by entering a hypothesized mean for each variable. The test assumes multivariate normality of the Y variables. The Hotelling’s T2 Test report gives the following: Variable Lists the variables entered as Y. Mean Gives the sample mean for each variable. Hypothesized Mean Shows the null hypothesis means that you specified. Test Statistic Gives the value of Hotelling’s T2 statistic. F Ratio Gives the value of the test statistic. If you have n rows and k variables, the F ratio is given as follows: Prob > F The p-value for the test. Under the null hypothesis the F ratio has an F distribution with n and n - k degrees of freedom. Simple Statistics This menu contains two options that each show or hide simple statistics (mean, standard deviation, and so on) for each column. The univariate and multivariate simple statistics can differ when there are missing values present, or when the Robust method is used. Univariate Simple Statistics Shows statistics that are calculated on each column, regardless of values in other columns. These values match those produced by the Distribution platform. Multivariate Simple Statistics Shows statistics that correspond to the estimation method selected in the launch window. If the REML, ML, or Robust method is selected, the mean vector and covariance matrix are estimated by that selected method. If the Row-wise method is selected, all rows with at least one missing value are excluded from the calculation of means and variances. If the Pairwise method is selected, the mean and variance are calculated for each column. These options are off by default. Nonparametric Correlations This menu contains three nonparametric measures: Spearman’s Rho, Kendall’s Tau, and Hoeffding’s D. These options are off by default. Set α Level You can specify any alpha value for the correlation confidence intervals. Four alpha values are listed: 0.01, 0.05, 0.10, and 0.50. Select Other to enter any other value. Scatterplot Matrix Shows or hides a scatterplot matrix of each pair of response variables. This option is on by default. Color Maps The Color Map menu contains three types of color maps. Color Map On Correlations Produces a cell plot that shows the correlations among variables on a scale from red (+1) to blue (-1). Color Map On p-values Produces a cell plot that shows the significance of the correlations on a scale from p = 0 (red) to p = 1 (blue). Cluster the Correlations Produces a cell plot that clusters together similar variables. The correlations are the same as for Color Map on Correlations, but the positioning of the variables may be different. These options are off by default. Parallel Coord Plot Shows or hides a parallel coordinate plot of the variables. This option is off by default. Ellipsoid 3D Plot Shows or hides a 95% confidence ellipsoid around three variables that you are asked to specify the three variables. This option is off by default. Principal Components Principal components is a technique to take linear combinations of the original variables. The first principal component has maximum variation, the second principal component has the next most variation, subject to being orthogonal to the first, and so on. For details, see the chapter Principal Components. Outlier Analysis Item Reliability This menu contains options that each shows or hides an item reliability report. The reports indicate how consistently a set of instruments measures an overall response, using either Cronbach’s α or standardized α. These options are off by default. Impute Missing Data Produces a new data table that duplicates your data table and replaces all missing values with estimated values. This option is available only if your data table contains missing values. Save Imputed Formula Script
The Nonparametric Correlations menu offers three nonparametric measures:
is based on the number of concordant and discordant pairs of observations. A pair is concordant if the observation with the larger value of X also has the larger value of Y. A pair is discordant if the observation with the larger value of X has the smaller value of Y. There is a correction for tied pairs (pairs of observations that have equal values of X or equal values of Y).
Clusters of Correlations
When you look for patterns in the scatterplot matrix, you can see the variables cluster into groups based on their correlations. Clusters of Correlations shows two clusters of correlations: the first two variables (top, left), and the next four (bottom, right).
 Show Points Shows or hides the points in the scatterplots. Fit Line Shows or hides the regression line and 95% level confidence curves for the fitted regression line. Density Ellipses Shows or hides the 95% density ellipses in the scatterplots. Use the Ellipse α menu to change the α-level. Shaded Ellipses Colors each ellipse. Use the Ellipses Transparency and Ellipse Color menus to change the transparency and color. Show Correlations Shows or hides the correlation of each histogram in the upper left corner of each scatterplot. Show Histogram Shows either horizontal or vertical histograms in the label cells. Once histograms have been added, select Show Counts to label each bar of the histogram with its count. Select Horizontal or Vertical to either change the orientation of the histograms or remove the histograms. Sets the α-level used for the ellipses. Select one of the standard α-levels in the menu, or select Other to enter a different one. Ellipses Transparency Sets the transparency of the ellipses if they are colored. Select one of the default levels, or select Other to enter a different one. The default value is 0.2. Ellipse Color Sets the color of the ellipses if they are colored. Select one of the colors in the palette, or select Other to use another color. The default value is red. Nonpar Density Shows or hides shaded density contours based on a smooth nonparametric bivariate surface that describes the density of data points. Contours for the 10% and 50% quantiles of the nonparametric surface are shown.
The Outlier Analysis menu contains options that show or hide plots that measure distance in the multivariate sense using one of these methods:
 • Mahalanobis distance
 • jackknife distances
 • T2 statistic
In Example of an Outlier, Point A is an outlier because it is outside the correlation structure rather than because it is an outlier in any of the coordinate directions.
Example of an Outlier
T2 Statistic
The T2 plot shows distances that are the square of the Mahalanobis distance. This plot is preferred for multivariate control charts. The plot includes the value of the calculated T2 statistic, as well as its upper control limit. Values that fall outside this limit might be outliers. See T2 Distance Measures for more information.
You can save any of the distances to the data table by selecting the Save option from the red triangle menu for the plot.
Note: There is no formula saved with the jackknife distance column. This means that the distance is not recomputed if you modify the data table. If you add or delete columns, or change values in the data table, select Analyze > Multivariate Methods > Multivariate again to compute new jackknife distances.
Item reliability indicates how consistently a set of instruments measures an overall response. Cronbach’s α (Cronbach 1951) is one measure of reliability. Two primary applications for Cronbach’s α are industrial instrument reliability and questionnaire analysis.
Cronbach’s α is based on the average correlation of items in a measurement scale. It is equivalent to computing the average of all split-half correlations in the data table. The Standardized α can be requested if the items have variances that vary widely.
Note: Cronbach’s α is not related to a significance level α. Also, item reliability is unrelated to survival time reliability analysis.
To look at the influence of an individual item, JMP excludes it from the computations and shows the effect of the Cronbach’s α value. If α increases when you exclude a variable (item), that variable is not highly correlated with the other variables. If the α decreases, you can conclude that the variable is correlated with the other items in the scale. Nunnally (1979) suggests a Cronbach’s α of 0.7 as a rule-of-thumb acceptable level of agreement.
See Cronbach’s α for details about computations.
To impute missing data, select Impute Missing Data from the red triangle menu for Multivariate. A new data table is created that duplicates your data table and replaces all missing values with estimated values.