Example 44.1: Variability of Calcium Concentration in Turnip Greens

The NESTED Procedure

Example 44.1: Variability of Calcium Concentration in Turnip Greens

In the following example from Snedecor and Cochran (1976), an experiment is conducted to study the variability of calcium concentration in turnip greens. Four plants are selected at random; then three leaves are randomly selected from each plant. Two 100-mg samples are taken from each leaf. The amount of calcium is determined by microchemical methods.

Because the data are read in sorted order, it is not necessary to use PROC SORT on the CLASS variables. Leaf is nested in Plant; Sample is nested in Leaf and is left for the residual term. All the effects are random effects. The following statements read the data and invoke PROC NESTED. These statements produce Output 44.1.1:

   title 'Calcium Concentration in Turnip Leaves'
         '--Nested Random Model';
   title2 'Snedecor and Cochran, ''Statistical Methods'''
         ', 1976, p. 286';
   data Turnip;
      do Plant=1 to 4;
         do Leaf=1 to 3;
            do Sample=1 to 2;
               input Calcium @@;
               output;
               end;
            end;
         end;

      datalines;
   3.28 3.09 3.52 3.48 2.88 2.80
   2.46 2.44 1.87 1.92 2.19 2.19
   2.77 2.66 3.74 3.44 2.55 2.55
   3.78 3.87 4.07 4.12 3.31 3.31
   ;
   proc nested;
      class Plant Leaf;
      var Calcium;
   run;

Output 44.1.1: Analysis of Calcium Concentration in Turnip Greens Using PROC NESTED

Calcium Concentration in Turnip Leaves--Nested Random Model

Snedecor and Cochran, 'Statistical Methods', 1976, p. 286

The NESTED Procedure

Coefficients of Expected Mean Squares
Source	Plant	Leaf	Error
Plant	6	2	1
Leaf	0	2	1
Error	0	0	1

Calcium Concentration in Turnip Leaves--Nested Random Model

Snedecor and Cochran, 'Statistical Methods', 1976, p. 286

The NESTED Procedure

Nested Random Effects Analysis of Variance for Variable Calcium
Variance Source	DF	Sum of Squares	F Value	Pr > F	Error Term	Mean Square	Variance Component	Percent of Total
Total	23	10.270396				0.446539	0.532938	100.0000
Plant	3	7.560346	7.67	0.0097	Leaf	2.520115	0.365223	68.5302
Leaf	8	2.630200	49.41	<.0001	Error	0.328775	0.161060	30.2212
Error	12	0.079850				0.006654	0.006654	1.2486

Calcium Mean	3.01208333
Standard Error of Calcium Mean	0.32404445

The results indicate that there is significant (nonzero) variation from plant to plant (Pr > F is 0.0097) and from leaf to leaf within a plant (Pr > F is less than 0.0001). Notice that the variance component for Plant uses the Leaf mean square as an error term in the model rather than the error mean square.

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