Improve code to calculate Cronbach's alpha (don't use fixed numbers f…

…or counting variables) and provide an alternative method of calculation (some should be more “intuitive”)

Factor analysis/Cronbach.R

@@ -1,24 +1,41 @@
-# Cronbach's alpha sets the inter-item covariances into relation to the
-# total amount of variation (i.e., a sum of the inter-item covariances
-# and the item variances)
+# Cronbach's alpha sets the inter-item covariances into relation to the total
+# amount of variation (i.e., a sum of the inter-item covariances and the item
+# variances)
 
-# we use the covariance matrix (cov()) for this calculation: it contains
-# the coviarances (outside the main diagonal) and the variances in the
-# main diagonal
-X = cov(data[2:6])
+# we use the covariance matrix (cov()) for this calculation: it contains the
+# coviarances (outside the main diagonal) and the variances in the main
+# diagonal
+# adjust the columns [2:6] to match the columns / variables included in your
+# Reliability Analysis
+X <- cov(data[2:6], use = "complete.obs")
 X
 
 # the variances indicate how much any item varies by itself, the
 # covariances indicate how much the items vary together
 # we first consider the covariances (i.e., their mean)
 mean(abs(X[lower.tri(X)]))
 # this values is then multiplied with the number of values in the
-# covariance matrix (5 variables x 5 variables)
-mean(abs(X[lower.tri(X)])) * 5 ^ 2
+# covariance matrix (number of rows x number of columns)
+mean(abs(X[lower.tri(X)])) * nrow(X) * ncols(X)
 # and that values is then set in relation to (i.e., divided by) the total
 # amount of variation, we first show the value for the total amount of
 # variation
 sum(abs(X))
 # and then we set the inter-item covariances (their mean multiplied by the
 # number of possible combination) into relation to the total variation
-mean(abs(X[lower.tri(X)])) * 5 ^ 2 / sum(abs(X))
+mean(abs(X[lower.tri(X)])) * nrow(X) * ncols(X) / sum(abs(X))
+
+# -----------------------------------------------------------------------------
+
+# another way to calculate Cronbach's alpha that may be more “intuitive” is:
+# we take the variance-covariance-matrix (X) and copy it to a new variable (Y)
+Y <- X
+# in that new variable, we replace the values in the main diagonal of Y (the
+# variances of each of the variables) with the average value of the
+# inter-item-covariances (ie., how much the items in the questionnaire on
+# average covary with one another)
+diag(Y) <- mean(abs(X[lower.tri(X)]))
+
+# we divide that covariance matrix Y (where we replaced the variances by the
+# average inter-item covariances) by the original variance-covariance-matrix (X)
+sum(abs(Y)) / sum(abs(X))