Example

Example 49.1. 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 49.1.1:

  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 49.1.1: 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                Sum of                   Error                    Variance  Percent   Source        DF       Squares  F Value  Pr > F  Term    Mean Square     Component  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.