

Example 2.5 Computing Cronbach’s Coefficient Alpha
The following statements create the data set Fish1 from the Fish data set used in
Chapter 83,
The STEPDISC Procedure
(SAS/STAT 9.22 User's Guide). The cubic root of the weight (Weight3) is computed as a onedimensional measure of the size of a fish.
* Fish Measurement Data *
 The data set contains 35 fish from the species Bream caught in 
 Finland's lake Laengelmavesi with the following measurements: 
 Weight (in grams) 
 Length3 (length from the nose to the end of its tail, in cm) 
 HtPct (max height, as percentage of Length3) 
 WidthPct (max width, as percentage of Length3) 
**;
data Fish1 (drop=HtPct WidthPct);
title 'Fish Measurement Data';
input Weight Length3 HtPct WidthPct @@;
Weight3= Weight**(1/3);
Height=HtPct*Length3/100;
Width=WidthPct*Length3/100;
datalines;
242.0 30.0 38.4 13.4 290.0 31.2 40.0 13.8
340.0 31.1 39.8 15.1 363.0 33.5 38.0 13.3
430.0 34.0 36.6 15.1 450.0 34.7 39.2 14.2
500.0 34.5 41.1 15.3 390.0 35.0 36.2 13.4
450.0 35.1 39.9 13.8 500.0 36.2 39.3 13.7
475.0 36.2 39.4 14.1 500.0 36.2 39.7 13.3
500.0 36.4 37.8 12.0 . 37.3 37.3 13.6
600.0 37.2 40.2 13.9 600.0 37.2 41.5 15.0
700.0 38.3 38.8 13.8 700.0 38.5 38.8 13.5
610.0 38.6 40.5 13.3 650.0 38.7 37.4 14.8
575.0 39.5 38.3 14.1 685.0 39.2 40.8 13.7
620.0 39.7 39.1 13.3 680.0 40.6 38.1 15.1
700.0 40.5 40.1 13.8 725.0 40.9 40.0 14.8
720.0 40.6 40.3 15.0 714.0 41.5 39.8 14.1
850.0 41.6 40.6 14.9 1000.0 42.6 44.5 15.5
920.0 44.1 40.9 14.3 955.0 44.0 41.1 14.3
925.0 45.3 41.4 14.9 975.0 45.9 40.6 14.7
950.0 46.5 37.9 13.7
;
The following statements request a correlation analysis and compute Cronbach’s coefficient alpha for the variables Weight3, Length3, Height, and Width:
ods graphics on;
title 'Fish Measurement Data';
proc corr data=fish1 nomiss alpha plots;
var Weight3 Length3 Height Width;
run;
ods graphics off;
The NOMISS option excludes observations with missing values, and the ALPHA option computes Cronbach’s coefficient alpha for the analysis variables.
The "Simple Statistics" table in Output 2.5.1 displays univariate descriptive statistics for each analysis variable.
Output 2.5.1
Simple Statistics
Weight3 Length3 Height Width 
34 
8.44751 
0.97574 
287.21524 
6.23168 
10.00000 
34 
38.38529 
4.21628 
1305 
30.00000 
46.50000 
34 
15.22057 
1.98159 
517.49950 
11.52000 
18.95700 
34 
5.43805 
0.72967 
184.89370 
4.02000 
6.74970 
The "Pearson Correlation Coefficients" table in Output 2.5.2 displays Pearson correlation statistics for pairs of analysis variables. When you specify the NOMISS option, the same set of 34 observations is used to compute the correlation for each pair of variables.
Output 2.5.2
Pearson Correlation Coefficients
Since the data set contains only one species of fish, all the variables are highly correlated. Using the ALPHA option, the CORR procedure computes Cronbach’s coefficient alpha in Output 2.5.3. The Cronbach’s coefficient alpha is a lower bound for the reliability coefficient for the raw variables and the standardized variables. Positive correlation is needed for the alpha coefficient because variables measure a common entity.
Output 2.5.3
Cronbach’s Coefficient Alpha
Raw 
0.822134 
Standardized 
0.985145 
Because the variances of some variables vary widely, you should use the standardized score to estimate reliability. The overall standardized Cronbach’s coefficient alpha of 0.985145 provides an acceptable lower bound for the reliability coefficient. This is much greater than the suggested value of 0.70 given by Nunnally and Bernstein (1994).
The standardized alpha coefficient provides information about how each variable reflects the reliability of the scale with standardized variables. If the standardized alpha decreases after removing a variable from the construct, then this variable is strongly correlated with other variables in the scale. On the other hand, if the standardized alpha increases after removing a variable from the construct, then removing this variable from the scale makes the construct more reliable. The "Cronbach Coefficient Alpha with Deleted Variables" table in Output 2.5.4 does not show significant increase or decrease in the standardized alpha coefficients. See the section Cronbach’s Coefficient Alpha for more information about Cronbach’s alpha.
Output 2.5.4
Cronbach’s Coefficient Alpha with Deleted Variables
0.975379 
0.783365 
0.973464 
0.977103 
0.967602 
0.881987 
0.967177 
0.978783 
0.964715 
0.655098 
0.968079 
0.978542 
0.934635 
0.824069 
0.937599 
0.986626 
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