Example 7.18 Design and Analysis of a Complete Factorial Experiment

Note: See Complete Factorial Experiment in the SAS/QC Sample Library.

Yin and Jillie (1987) describe an experiment on a nitride etch process for a single-wafer plasma etcher. The experiment has four factors: cathode power (Power), gas flow (Flow), reactor chamber pressure (Pressure), and electrode gap (Gap). A single replicate of a $2^4$ design is run, and the etch rate (Rate) is measured. You can use the following statements to construct a 16-run design in the four factors:

proc factex;
   factors Power Flow Pressure Gap;
   output out=EtcherDesign
          Power    nvals=(0.80 1.20)
          Flow     nvals=(4.50 550)
          Pressure nvals=(125  200)
          Gap      nvals=(275  325);
run;

The design with the actual (decoded) factor levels is saved in the data set EtcherDesign. The experiment that uses the 16-run design is performed, and the etch rate is measured. The following DATA step updates the data set EtcherDesign with the values of Rate:

data EtcherDesign;
   set EtcherDesign;
   input Rate @@;
   datalines;
   550   669  604   650  633   642  601   635
   1037  749  1052  868  1075  860  1063  729
   ;
title 'Nitride Etch Process Experiment';
proc print;
run;

The data set DESGNDAT is listed in Output 7.18.1.

Output 7.18.1: A $2^4$ Design with Responses

Nitride Etch Process Experiment

Obs Power Flow Pressure Gap Rate
1 0.8 4.5 125 275 550
2 0.8 4.5 125 325 669
3 0.8 4.5 200 275 604
4 0.8 4.5 200 325 650
5 0.8 550.0 125 275 633
6 0.8 550.0 125 325 642
7 0.8 550.0 200 275 601
8 0.8 550.0 200 325 635
9 1.2 4.5 125 275 1037
10 1.2 4.5 125 325 749
11 1.2 4.5 200 275 1052
12 1.2 4.5 200 325 868
13 1.2 550.0 125 275 1075
14 1.2 550.0 125 325 860
15 1.2 550.0 200 275 1063
16 1.2 550.0 200 325 729


To perform an analysis of variance on the responses, you can use the GLM procedure, as follows:

proc glm data=EtcherDesign;
   class Power Flow Pressure Gap;
   model rate=Power|Flow|Pressure|Gap@2 / ss1;
run;

The factors are listed in both the CLASS and MODEL statements, and the response as a function of the factors is modeled by using the MODEL statement. The MODEL statement requests Type I sum of squares (SS1) and lists all effects that contain two or fewer factors. It is assumed that three-factor and higher interactions are not significant.

Part of the output from the GLM procedure is shown in Output 7.18.2. The main effect of the factors Power and Gap and the interaction between Power and Gap are significant (their p-values are less than 0.01).

Output 7.18.2: Analysis of Variance for the Nitride Etch Process Experiment

Nitride Etch Process Experiment

The GLM Procedure
 
Dependent Variable: Rate

Source DF Type I SS Mean Square F Value Pr > F
Power 1 374850.0625 374850.0625 183.99 <.0001
Flow 1 217.5625 217.5625 0.11 0.7571
Power*Flow 1 18.0625 18.0625 0.01 0.9286
Pressure 1 10.5625 10.5625 0.01 0.9454
Power*Pressure 1 1.5625 1.5625 0.00 0.9790
Flow*Pressure 1 7700.0625 7700.0625 3.78 0.1095
Gap 1 41310.5625 41310.5625 20.28 0.0064
Power*Gap 1 94402.5625 94402.5625 46.34 0.0010
Flow*Gap 1 2475.0625 2475.0625 1.21 0.3206
Pressure*Gap 1 248.0625 248.0625 0.12 0.7414