IE 5342-001 Assignment 17

1 Factor of Yield

a. Half normal plot

library(FrF2)
dat <- data.frame(
  A = c(-1, 1, -1, 1, -1, 1, -1, 1,
        -1, 1, -1, 1, -1, 1, -1, 1),
  
  B = c(-1, -1, 1, 1, -1, -1, 1, 1,
        -1, -1, 1, 1, -1, -1, 1, 1),
  
  C = c(-1, -1, -1, -1, 1, 1, 1, 1,
        -1, -1, -1, -1, 1, 1, 1, 1),
  
  D = c(-1, -1, -1, -1, -1, -1, -1, -1,
        1, 1, 1, 1, 1, 1, 1, 1),
  
  Yield = c(12, 18, 13, 16, 17, 15, 20, 15,
            10, 25, 13, 24, 19, 21, 17, 23)
)

model <- lm(Yield ~ A * B * C * D, data = dat)
halfnormal(model)

Based on the half-normal plot, factors A and D and the interactions AC and AD appear to have the largest effects on Yield.

b. Significant terms

We will retain A, C, D, AC, and AD. Although C is not among the four largest effects, we retain it because the AC interaction is important.

reduced_model <- lm(
  Yield ~ A + C + D + A:C + A:D,
  data = dat
)

anova(reduced_model)

## Analysis of Variance Table
## 
## Response: Yield
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## A          1  81.00  81.000 49.8462 3.456e-05 ***
## C          1  16.00  16.000  9.8462 0.0105485 *  
## D          1  42.25  42.250 26.0000 0.0004647 ***
## A:C        1  72.25  72.250 44.4615 5.583e-05 ***
## A:D        1  64.00  64.000 39.3846 9.193e-05 ***
## Residuals 10  16.25   1.625                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The smaller effects were pooled into error. Factor C was retained to maintain model hierarchy because the AC interaction was included. The reduced ANOVA model shows that A, C, D, AC, and AD are significant at the 0.05 level of significance, with P-values of 0.000035, 0.010549, 0.000465, 0.000056, and 0.000092, respectively. Therefore, these effects are retained in the final model.