Factorial Designs

STA6246

Author

Dr. Cohen

Battery Life Experiment

  • An experimenter wanted to study the life of a battery in a the design phase.

  • The engineer deicides to test a=3 plate materials and b=3 temperature (15, 75, and 125) degrees F.

  • n=4 batteries were used

  • N=abn = 36

Data description

library(tidyverse)
library(gtsummary)

Lifebattery=c(130,155,74,180,150,188,126,159,138,110,168,160,34,40,75,80,136,122,106,115,174,120,150,139,82,20,70,58,25,70,58,45,96,104,60,82)

MaterialType= gl(n = 3,k=4,length = 36,labels = (1:3))

Temp= gl(n = 3,k=12,length = 36,labels = c(15,70,125))

battery_data =tibble(Lifebattery,MaterialType,Temp)

## Plotting

interaction.plot(x.factor = Temp,trace.factor = MaterialType,response = Lifebattery,fun = mean)

head(battery_data)
# A tibble: 6 × 3
  Lifebattery MaterialType Temp 
        <dbl> <fct>        <fct>
1         130 1            15   
2         155 1            15   
3          74 1            15   
4         180 1            15   
5         150 2            15   
6         188 2            15   
battery_data %>%
  ggplot(aes(x=Temp,y=Lifebattery,group=Temp)) +
  geom_point()+
  stat_summary(aes(group=1),
               fun = mean, 
               geom = "line", 
               col="red",
               linewidth=2)+
  stat_summary(aes(label=round(..y..,1),group=1),
               fun = mean, 
               geom = "text", 
               col="blue",
               size=10)+
  theme_bw()+
  labs(x="Temp",
       y="Battery Life (in hours)")+
  facet_wrap(~MaterialType)

battery_data %>% 
  select(Lifebattery,Temp) %>%
  tbl_summary(by=Temp,
              statistic = list(all_continuous() ~ "{mean} ({sd})"))
Characteristic 15, N = 121 70, N = 121 125, N = 121
Lifebattery 145 (32) 108 (43) 64 (26)
1 Mean (SD)

  • The mean battery life is reduced when the temperature increases for all 3 material types.

  • add more comments

ANOVA in R

Fitting
library(sjPlot)

# ANOVA one factor and one blocking factor
results=aov(Lifebattery~ MaterialType *Temp)
summary(results)
                  Df Sum Sq Mean Sq F value   Pr(>F)    
MaterialType       2  10684    5342   7.911  0.00198 ** 
Temp               2  39119   19559  28.968 1.91e-07 ***
MaterialType:Temp  4   9614    2403   3.560  0.01861 *  
Residuals         27  18231     675                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Individual CIs 
confint(results)
                           2.5 %    97.5 %
(Intercept)            108.09174 161.40826
MaterialType2          -16.70048  58.70048
MaterialType3          -28.45048  46.95048
Temp70                -115.20048 -39.79952
Temp125               -114.95048 -39.54952
MaterialType2:Temp70   -11.81653  94.81653
MaterialType3:Temp70    25.93347 132.56653
MaterialType2:Temp125  -82.31653  24.31653
MaterialType3:Temp125  -34.56653  72.06653

  • There was a significance main effect of the temperature on the battery life (p=0.00198)

  • There was a significance main effect of the Material Type on the battery life (p<0.001)

  • There was a significance interaction effect of the temperature and Material Type on the battery life (p=0.0186)

Model Adequacy
par(mfrow=c(2,2))
plot(results)

No transformation is needed.

Multiple Comparison

Considering 5% level of significance.

#Tukey Test 
THSD=TukeyHSD(results,which = "MaterialType")
plot(THSD,las=1)

THSD
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = Lifebattery ~ MaterialType * Temp)

$MaterialType
        diff       lwr      upr     p adj
2-1 25.16667 -1.135677 51.46901 0.0627571
3-1 41.91667 15.614323 68.21901 0.0014162
3-2 16.75000 -9.552344 43.05234 0.2717815
# Fisher LSD Test
library(agricolae)
L=LSD.test(results,"MaterialType",console=TRUE,group = F)

Study: results ~ "MaterialType"

LSD t Test for Lifebattery 

Mean Square Error:  675.213 

MaterialType,  means and individual ( 95 %) CI

  Lifebattery      std  r       LCL       UCL Min Max
1    83.16667 48.58888 12  67.77551  98.55782  20 180
2   108.33333 49.47237 12  92.94218 123.72449  25 188
3   125.08333 35.76555 12 109.69218 140.47449  60 174

Alpha: 0.05 ; DF Error: 27
Critical Value of t: 2.051831 

Comparison between treatments means

      difference pvalue signif.       LCL        UCL
1 - 2  -25.16667 0.0251       * -46.93305  -3.400285
1 - 3  -41.91667 0.0005     *** -63.68305 -20.150285
2 - 3  -16.75000 0.1260         -38.51638   5.016382

Conclusion:

  • Tukey’s Test indicates:

    • Significant differences between:

      • all temperatures

      • Material Type 1 and 3

  • Fisher LSD Test indicates:

    • Significant differences between:

      • all temperatures

      • Material Type 1 and 3

      • Material Type 1 and 2

The material 3 seems to be robust to higher and lower temperature.

The material 2 seems to be better fitted to lower temperature.

Impurity data (One cell observation)

library(additivityTests)

impurity=c(5,3,1,4,1,1,6,4,3,3,2,1,5,3,2)

Press= gl(n = 5,k=3,length = 15,labels = seq(25,45,5))

Temp= gl(n = 3,k=1,length = 15,labels = c(100,125,150))

data=matrix(impurity,nrow = 3)

# additivity Test (no-interaction)
tukey.test(data)

Tukey test on 5% alpha-level:

Test statistic: 0.3627 
Critival value: 5.591 
The additivity hypothesis cannot be rejected.
av=aov(impurity ~ Press + Temp)
summary(av)
            Df Sum Sq Mean Sq F value   Pr(>F)    
Press        4  11.60    2.90   11.60  0.00206 ** 
Temp         2  23.33   11.67   46.67 3.88e-05 ***
Residuals    8   2.00    0.25                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Radar Scope detection

2 factors: 3x2 factorial design.

An engineer is studying methods for improving the ability to detect targets on a radar scope.

Two factors she considers to be important are the amount of background noise, or “ground clutter,” on the scope and the type of filter placed over the screen.

An experiment is designed using three levels of ground clutter and two filter types.

We will consider these as fixed‐type factors.

The experiment is performed by randomly selecting a treatment combination (ground clutter level and filter type) and then introducing a signal representing the target into the scope.

The intensity of this target is increased until the operator observes it.

The intensity level at detection is then measured as the response variable.

Blocking

Because of operator availability, it is convenient to select an operator and keep him or her at the scope until all the necessary runs have been made. Furthermore, operators differ in their skill and ability to use the scope. Consequently, it seems logical to use the operators as blocks. Four operators are randomly selected.

intensity=c(90,102,114,86,87,93,96,106,112,84,90,91,100,105,108,92,97,95,92,96,98,81,80,83)

Noise= gl(n = 3,k=1,length = 24,labels = (1:3))
Noise
 [1] 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Levels: 1 2 3
Fil=gl(n = 2,k=3,length = 24,labels = (1:2))

Op=gl(n = 4,k=6,length = 24,labels = (1:4))

av=aov(intensity ~ Noise * Fil + Op)


summary(av)
            Df Sum Sq Mean Sq F value   Pr(>F)    
Noise        2  335.6   167.8  15.132 0.000253 ***
Fil          1 1066.7  1066.7  96.192 6.45e-08 ***
Op           3  402.2   134.1  12.089 0.000277 ***
Noise:Fil    2   77.1    38.5   3.476 0.057507 .  
Residuals   15  166.3    11.1                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Random Effect
library(lme4)
avRE <- lmer(intensity ~ (1 | Op) +Noise*Fil)
summary(avRE)
Linear mixed model fit by REML ['lmerMod']
Formula: intensity ~ (1 | Op) + Noise * Fil

REML criterion at convergence: 110.2

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.4661 -0.4704 -0.0127  0.6198  1.6870 

Random effects:
 Groups   Name        Variance Std.Dev.
 Op       (Intercept) 20.49    4.527   
 Residual             11.09    3.330   
Number of obs: 24, groups:  Op, 4

Fixed effects:
            Estimate Std. Error t value
(Intercept)   94.500      2.810  33.630
Noise2         7.750      2.355   3.291
Noise3        13.500      2.355   5.733
Fil2          -8.750      2.355  -3.716
Noise2:Fil2   -5.000      3.330  -1.502
Noise3:Fil2   -8.750      3.330  -2.628

Correlation of Fixed Effects:
            (Intr) Noise2 Noise3 Fil2   Ns2:F2
Noise2      -0.419                            
Noise3      -0.419  0.500                     
Fil2        -0.419  0.500  0.500              
Noise2:Fil2  0.296 -0.707 -0.354 -0.707       
Noise3:Fil2  0.296 -0.354 -0.707 -0.707  0.500
# Confidence Intervals Variance Components and Overall Mean
confint(avRE,oldNames=FALSE)
                       2.5 %      97.5 %
sd_(Intercept)|Op   1.989122   9.9695167
sigma               2.177544   4.0740129
(Intercept)        88.882193 100.1178095
Noise2              3.553437  11.9465629
Noise3              9.303437  17.6965629
Fil2              -12.946563  -4.5534371
Noise2:Fil2       -10.934836   0.9348361
Noise3:Fil2       -14.684836  -2.8151639