STAT 3220 Unit 4.3-4.4

4.3 One Factor ANOVA

A sample of 44 healthy male college students participated in an experiment, Each student was asked to memorize a list of 40 words (20 on a green list and 20 on a red list).The students were then randomly assigned of one of 4 treatments: Group A: received 2 alcoholic drinks, Group AC: received 2 alcoholic drinks with caffeine powder dissolved in their drinks, Group AR: received 2 alcoholic drinks and a monetary reward for correct responses on the task, Group P: were told they received 2 alcoholic drinks, but instead just received a carbonated drink.

After consuming their drinks and resting for 25 minutes, the students performed the word completion task. The response variable ”task score” represents the difference between the proportion of correct responses on the green list and incorrect on the red list.

Step 1: Collect the Data and Determine Experimental Design

Consider the design of the data collection: For the One and Two Factor ANOVA we are covering the analysis on completely randomized designs. We want to ensure our data meet that criteria and define the other components of the experimental design.

drinkersdata<-read.delim("https://raw.githubusercontent.com/kvaranyak4/STAT3220/main/DRINKERS.txt",header=T)
names(drinkersdata)

[1] "GROUP" "SCORE"

head(drinkersdata)

table(drinkersdata$GROUP)

 A AC AR  P 
11 11 11 11 

Experimental design

This is a completely randomized design because subjects are randomly assigned to treatments. There is one factor, group, with four levels: AR, AC, A, and P. Therefore, there are four treatments: AR, AC, A, P. It is balanced because each treatment has the same number of experimental units. The response variable is the task score.

Step 2: Hypothesize Relationship (Exploratory Data Analysis)

We will look at side by side box plots of the treatment groups and compare the mean value. We can also compare the 5 number summary for a more detailed interpretation. (We could also determine the approximate distribution of each treatment group- in the next section we will see that we want the treatments to have normally distributed data.)

boxplot(SCORE~GROUP, data=drinkersdata)

tapply(drinkersdata$SCORE,drinkersdata$GROUP,summary)

$A
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-0.35000 -0.05000  0.16000  0.06364  0.19000  0.31000 

$AC
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0200  0.1750  0.2200  0.2655  0.3750  0.5000 

$AR
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  0.000   0.400   0.500   0.440   0.525   0.610 

$P
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   0.12    0.33    0.43    0.40    0.47    0.62 

• It appears there are differences in the mean task scores for the four groups, so we will perform a statistical analysis to compare the means through the ANOVA procedure.

Step 3: Perform the ANOVA

• Null Hypothesis \(H_0 : \mu_A = \mu_{AC} = \mu_{AR} = \mu_P\)
• Alternative Hypothesis at least two means are different

drinkanova1<-aov(SCORE~GROUP, data=drinkersdata)
summary(drinkanova1)

            Df Sum Sq Mean Sq F value   Pr(>F)    
GROUP        3 0.9506  0.3169   10.29 3.76e-05 ***
Residuals   40 1.2317  0.0308                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Test Statistic F test: F = MST/MSE = 0.317/0.0308 = 10.29
• Distribution of Test Statistic : F with numerator degrees of freedom, v1 = 4 − 1 and denominator degrees of freedom, v2 = 44 − 4
• CONCLUSION: With a pvalue this small, we reject the null hypothesis. There is evidence that at least two population treatment means are different from each other. Based on the sample means, it appears the group with just alcohol has a much lower than the rest, while the group with a monetary reward had a much higher mean than the rest. We will perform the post-hoc test to compare pairwise means in the next section.

Contextual Conclusion

The question of interest is: Does coffee or some other form of stimulation really allow a person suffering from alcohol intoxication to ”sober up”? What is your conclusion? Include any relevant inferences and confidence intervals. Use alpha=0.05.

TukeyHSD(drinkanova1,conf.level=.95)

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

Fit: aov(formula = SCORE ~ GROUP, data = drinkersdata)

$GROUP
            diff          lwr       upr     p adj
AC-A   0.2018182  0.001256169 0.4023802 0.0480809
AR-A   0.3763636  0.175801623 0.5769256 0.0000618
P-A    0.3363636  0.135801623 0.5369256 0.0003282
AR-AC  0.1745455 -0.026016558 0.3751075 0.1075832
P-AC   0.1345455 -0.066016558 0.3351075 0.2892292
P-AR  -0.0400000 -0.240562013 0.1605620 0.9501001

plot(TukeyHSD(drinkanova1,conf.level=.95))

Note: If 0 is captured in any interval, it would imply the means are not significantly different (we will examine this precise pairwise analysis in the next section.)

We will look at the confidence intervals for the difference in the mean scores of the alcohol group compared to the others. Note all of the confidence intervals are positive. This implies that the alcohol group under performs compared to every other group.

Pair	Lower	Upper
AC-A	0.00126	0.4024
AR-A	0.1758	0.5769
P-A	0.1358	0.5369

We are 95% confident that the true mean difference of scores for those who also had caffeine is between 0.00126 and 0.40 higher than the group with just alcohol. There is a significant difference between the group with just alcohol and the group with caffeine, but it may not be practically significant.

Next steps

Since the main effect is significant, we go to post hoc analysis (Tukey’s test in this class).

We will also check for the assumptions (next section).

4.3 Two Factor ANOVA

The chemical element antimony is sometimes added to tin – lead solder to replace the more expensive tin and to reduce the cost of soldering. A factorial experiment was conducted to determine how antimony affects the strength of the tin–lead solder joint (Journal of Materials Science, May 1986). Tin–lead solder specimens were prepared using one of four possible cooling methods (water- quenched, WQ; oil-quenched, OQ; air-blown, AB; and furnace-cooled, FC) and with one of four possible amounts of antimony (0%, 3%, 5%, and 10%) added to the composition. Three solder joints were randomly assigned to each of the 4 × 4 = 16 treatments and the shear strength of each measured.

Step 1: Collect the Data and Determine Experimental Design

Consider the design of the data collection: For the One and Two Factor ANOVA we are covering the analysis on completely randomized designs. We want to ensure our data meet that criteria and define the other components of the experimental design.

tindata<-read.delim("https://raw.githubusercontent.com/kvaranyak4/STAT3220/main/TINLEAD.txt",header=T)
names(tindata)

[1] "ANTIMONY" "METHOD"   "STRENGTH"

head(tindata)

table(tindata$ANTIMONY,tindata$METHOD)

    
     AB FC OQ WQ
  0   3  3  3  3
  3   3  3  3  3
  5   3  3  3  3
  10  3  3  3  3

Experimental design

This is a completely randomized design because experimental units are randomly assigned to treatments. There two factors, cooling method, with four levels: AB, FC, OQ, and WQ and antimony with four levels 0%, 3%, 5%, and 10%. Therefore, there are 16 treatments. It is balanced because each treatment has the same number of experimental units. The response variable is the sheer strength.

Step 2: Hypothesize Relationship (Exploratory Data Analysis)

For two factor designs, we will examine the interaction plot first.

# We want to begin with examining the interaction
interaction.plot(tindata$METHOD, tindata$ANTIMONY, tindata$STRENGTH,fun=mean,trace.label="Antimony Level"
                 , xlab="Cooling Method",ylab="Mean Sheer Strength",main="Interaction Plot Cooling Method X Antimony")

# We can also look at the main effects
#cooling method
boxplot(STRENGTH~METHOD, data=tindata)

tapply(tindata$STRENGTH,tindata$METHOD,summary)

$AB
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  15.80   18.05   20.70   20.18   22.30   22.90 

$FC
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  16.40   17.60   19.50   18.95   20.00   20.90 

$OQ
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  16.40   19.75   20.50   20.45   21.15   24.30 

$WQ
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  15.20   17.48   18.80   18.64   19.50   22.30 

#antimony level
boxplot(STRENGTH~ANTIMONY, data=tindata)

tapply(tindata$STRENGTH,tindata$ANTIMONY,summary)

$`0`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  17.60   19.12   19.80   20.18   20.70   24.30 

$`3`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  18.60   19.38   20.20   20.41   21.10   22.90 

$`5`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  16.40   19.68   20.55   20.62   21.77   22.90 

$`10`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  15.20   16.40   17.10   17.02   17.60   19.00 

• INTERACTION: While there is some crossing in the interaction plot, there is not strong evidence of an interaction because generally the hierarchy stays the same across the levels. Additionally there are very small sample sizes in each treatment (3), so likely an interaction will not be significant.
• MAIN EFFECT (METHOD): The average value of the response appears similar across all groups.
• MAIN EFFECT (ANTIMONY): The average value of the response appears similar for 0%, 3%, and 5%, but different for 10%.

Step 3: Perform the ANOVA

Signifiance of Interaction

NOTE: With a two factor factorial design, we will always test the significance of the interaction first.

• Null Hypothesis \(H_0 : \mu_{AB,0} = \mu_{FC,0} = \mu_{OQ,0} = \mu_{WQ,0}=...=\mu_{WQ,10}\) (there are 16 means!)
• Alternative Hypothesis at least two means are different

# We include the interaction
# Because Antimony was read in as a numeric variable, we want to 
# indicate that it is actually a factor
tinanova1<-aov(STRENGTH~METHOD*factor(ANTIMONY), data=tindata)
summary(tinanova1)

                        Df Sum Sq Mean Sq F value   Pr(>F)    
METHOD                   3  28.63    9.54   5.527  0.00357 ** 
factor(ANTIMONY)         3 104.19   34.73  20.117 1.64e-07 ***
METHOD:factor(ANTIMONY)  9  25.13    2.79   1.617  0.15226    
Residuals               32  55.25    1.73                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Distribution of Test Statistic: F with (4-1)(4-1), 4*4(3-1)= 9, 32 DF
• Test Statistic F test: F = 1.62 with pvalue 0.15226
• CONCLUSION: With a pvalue this large, we fail to reject the null hypothesis. There is not evidence of an interaction. This means that the shear strength for each level of cooling method does not depend on the antimony level.
• Next step: remove the interaction and test the main effect

Significance of Main effects

#We do not include the interaction, just add the terms
# Because Antimony was read in as a numeric variable, we want to 
# indicate that it is actually a factor
tinanova2<-aov(STRENGTH~METHOD+factor(ANTIMONY), data=tindata)
summary(tinanova2)

                 Df Sum Sq Mean Sq F value   Pr(>F)    
METHOD            3  28.63    9.54   4.868  0.00549 ** 
factor(ANTIMONY)  3 104.19   34.73  17.716 1.58e-07 ***
Residuals        41  80.38    1.96                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Test for Method:
  • Null Hypothesis \(H_0 : \mu_{AB} = \mu_{FC} = \mu_{OQ} = \mu_{WQ}\)
  • Alternative Hypothesis at least two means are different
  • Test Statistic: 4.87 with p-value 0.0055
  • Conclusion: We can conclude at least one of the types of cooling methods produces a different shear strengths of the solders than the others.
• Test for Antimony:
  • Null Hypothesis \(H_0 : \mu_0 = \mu_3 = \mu_5 = \mu_{10}\)
  • Alternative Hypothesis at least two means are different
  • Test Statistic: 17.72 with p-value <.0001
  • Conclusion: We can conclude at least one of the levels of antimony produces a different shear strengths of the solders than the others.

Next steps

Since the main effects are significant, we go to post hoc analysis (Tukey’s test in this class).

We will also check for the assumptions (next section).

4.4 POST HOC + ASSUMPTIONS

Tin Example Continued

Step 4: Post Hoc Analysis (multiple comparisons test)

Since the main effects are significant, we go to post hoc analysis (Tukey’s test in this class).

TukeyHSD(tinanova2,conf.level = 0.95)

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

Fit: aov(formula = STRENGTH ~ METHOD + factor(ANTIMONY), data = tindata)

$METHOD
            diff         lwr          upr     p adj
FC-AB -1.2250000 -2.75555373  0.305553731 0.1567570
OQ-AB  0.2750000 -1.25555373  1.805553731 0.9628573
WQ-AB -1.5333333 -3.06388706 -0.002779602 0.0494355
OQ-FC  1.5000000 -0.03055373  3.030553731 0.0565864
WQ-FC -0.3083333 -1.83888706  1.222220398 0.9488521
WQ-OQ -1.8083333 -3.33888706 -0.277779602 0.0149509

$`factor(ANTIMONY)`
           diff       lwr       upr     p adj
3-0   0.2333333 -1.297220  1.763887 0.9767248
5-0   0.4416667 -1.088887  1.972220 0.8663147
10-0 -3.1583333 -4.688887 -1.627780 0.0000119
5-3   0.2083333 -1.322220  1.738887 0.9832100
10-3 -3.3916667 -4.922220 -1.861113 0.0000031
10-5 -3.6000000 -5.130554 -2.069446 0.0000010

plot(TukeyHSD(tinanova2,conf.level = 0.95))

• Method:
  • Water Quenched and Oil Quenched are the only pairwise means that are significantly different from each other. (WQ-AB have a pvalue very close to alpha- what does that mean?? We need to interpret pvalues!)
  • We are 95% confident that the Water Quenched solders have a shear strength between 0.277 and 3.339 lower than the oil quenched group.
• Antimony:
  • The 10% group is significantly different from the rest, but the other groups are not different from each other.
  • We are 95% confident that the 0% group has a shear strength between 1.628 and 4.688 higher than the 10% group.

Step 5: Assumptions

Normality

We would like to evaluate the distribution of each treatment.

hist(tindata$STRENGTH, xlab="Sheer Strength", main="Histogram of Response")

#NOTE: this function is qqPlot from the car package
qqPlot(tindata$STRENGTH, groups=tindata$METHOD)

qqPlot(tindata$STRENGTH, groups=tindata$ANTIMONY)

• The response variable appears approximately unimodal and symmetric as a whole and across each treatment of the main effects. The normality assumption is reasonably met.

Equal Variance

Because we determined the normality assumption was reasonably met, we will use Bartlett’s test of homogeneity.

#this function is in the stats package
bartlett.test(STRENGTH~ANTIMONY, data=tindata)

    Bartlett test of homogeneity of variances

data:  STRENGTH by ANTIMONY
Bartlett's K-squared = 5.0649, df = 3, p-value = 0.1671

bartlett.test(STRENGTH~METHOD, data=tindata)

    Bartlett test of homogeneity of variances

data:  STRENGTH by METHOD
Bartlett's K-squared = 2.6014, df = 3, p-value = 0.4572

• The null hypothesis is that the variance of each treatment is equal. With a p-value this large, we determine the equal variance assumption is met for both main effects.

Complete Example: Paper Towels

Three brands of paper towels were compared to test their strength. A random sample of 26 towels from each of the three brands were tested. Weights were added to the center of the towel as it was held tight. Then drops of water were added. There were three levels of water (0 drops, 5 drops, and 15 drops). The strength was recorded as how much weight the towel held before breaking.

Step 1: Collect the Data and Determine Experimental Design

Consider the design of the data collection: For the One and Two Factor ANOVA we are covering the analysis on completely randomized designs. We want to ensure our data meet that criteria and define the other components of the experimental design.

toweldata<-read.csv("https://raw.githubusercontent.com/kvaranyak4/STAT3220/main/C4%20PaperTowel2.csv",header=T)
names(toweldata)

[1] "Brand"    "Water"    "Strength"

head(toweldata)

table(toweldata$Brand,toweldata$Water)

           
             0  5 15
  Bounty    26 26 26
  Comfort   26 26 26
  Decorator 26 26 26

Experimental design

This is a completely randomized design because experimental units are randomly chosen from each treatment. There two factors, brand, with three levels: Bounty, Comfort, and Decorator and water with three levels 0, 5, and 15 drops. Therefore, there are 9 treatments. It is balanced because each treatment has the same number of experimental units. The response variable is the strength of the towel.

Step 2: Hypothesize Relationship (Exploratory Data Analysis)

For two factor designs, we will examine the interaction plot first. We can look at side by side box plots of the main effect treatment groups and compare the mean value.

# We want to begin with examining the interaction
interaction.plot(toweldata$Brand, toweldata$Water, toweldata$Strength,fun=mean,trace.label="Water Drops"
                 , xlab="Brand",ylab="Mean Strength",main="Interaction Plot Water Drops X Brand")

interaction.plot(toweldata$Water, toweldata$Brand, toweldata$Strength,fun=mean,trace.label="Brand"
                 , xlab="Water",ylab="Mean Strength",main="Interaction Plot Water Drops X Brand")

# We can also look at the main effects
#cooling method
boxplot(Strength~Brand, data=toweldata)

tapply(toweldata$Strength,toweldata$Brand,summary)

$Bounty
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1450    2050    2700    2599    3088    3600 

$Comfort
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  300.0   456.2  1700.0  1772.8  2900.0  3700.0 

$Decorator
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    300     500     700    1123    2000    2700 

#antimony level
boxplot(Strength~Water, data=toweldata)

tapply(toweldata$Strength,toweldata$Water,summary)

$`0`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1700    2425    2900    2824    3300    3700 

$`5`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    550     800    1700    1764    2650    3450 

$`15`
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  300.0   400.0   487.5   906.1  1687.5  2350.0 

Step 3: Perform the ANOVA

Significance of Interaction

NOTE: With a two factor factorial design, we will always test the significance of the interaction first.

• Null Hypothesis \(H_0 : \mu_{B,0} = \mu_{D,0} = \mu_{C,0} =...=\mu_{C,15}\) (there are 9 means!)
• Alternative Hypothesis at least two means are different

# We include the interaction
# Because Antimony was read in as a numeric variable, we want to 
# indicate that it is actually a factor

towelanova1<-aov(Strength~factor(Water)*Brand, data=toweldata)
summary(towelanova1)

                     Df    Sum Sq  Mean Sq F value Pr(>F)    
factor(Water)         2 144038884 72019442  1277.0 <2e-16 ***
Brand                 2  85291704 42645852   756.2 <2e-16 ***
factor(Water):Brand   4  27105924  6776481   120.2 <2e-16 ***
Residuals           225  12689471    56398                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

• Distribution of Test Statistic: F with (3-1)(3-1), 3*3(26-1)= 4,225 DF
• Test Statistic F test: F = 120.2 with pvalue <0.00001
• CONCLUSION: With a pvalue this small, we reject the null hypothesis. There is evidence of an interaction. This means that the strength for each level of brand does depend on the number of drops.
• It is appropriate to perform the post hoc analysis because the design is balanced and the interaction was significant.

Step 4: Post Hoc Analysis

TukeyHSD(towelanova1,conf.level = .95)

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

Fit: aov(formula = Strength ~ factor(Water) * Brand, data = toweldata)

$`factor(Water)`
           diff        lwr        upr p adj
5-0  -1059.9359 -1149.6554  -970.2164     0
15-0 -1918.2692 -2007.9887 -1828.5497     0
15-5  -858.3333  -948.0528  -768.6138     0

$Brand
                        diff        lwr        upr p adj
Comfort-Bounty     -825.9615  -915.6810  -736.2420     0
Decorator-Bounty  -1475.3205 -1565.0400 -1385.6010     0
Decorator-Comfort  -649.3590  -739.0785  -559.6395     0

$`factor(Water):Brand`
                                diff         lwr         upr     p adj
5:Bounty-0:Bounty         -171.15385  -377.46198    35.15428 0.1930999
15:Bounty-0:Bounty       -1176.92308 -1383.23121  -970.61495 0.0000000
0:Comfort-0:Bounty         157.69231   -48.61582   364.00044 0.2922960
5:Comfort-0:Bounty       -1336.53846 -1542.84659 -1130.23033 0.0000000
15:Comfort-0:Bounty      -2647.11538 -2853.42351 -2440.80725 0.0000000
0:Decorator-0:Bounty      -828.84615 -1035.15428  -622.53802 0.0000000
5:Decorator-0:Bounty     -2343.26923 -2549.57736 -2136.96110 0.0000000
15:Decorator-0:Bounty    -2601.92308 -2808.23121 -2395.61495 0.0000000
15:Bounty-5:Bounty       -1005.76923 -1212.07736  -799.46110 0.0000000
0:Comfort-5:Bounty         328.84615   122.53802   535.15428 0.0000416
5:Comfort-5:Bounty       -1165.38462 -1371.69275  -959.07649 0.0000000
15:Comfort-5:Bounty      -2475.96154 -2682.26967 -2269.65341 0.0000000
0:Decorator-5:Bounty      -657.69231  -864.00044  -451.38418 0.0000000
5:Decorator-5:Bounty     -2172.11538 -2378.42351 -1965.80725 0.0000000
15:Decorator-5:Bounty    -2430.76923 -2637.07736 -2224.46110 0.0000000
0:Comfort-15:Bounty       1334.61538  1128.30725  1540.92351 0.0000000
5:Comfort-15:Bounty       -159.61538  -365.92351    46.69275 0.2765142
15:Comfort-15:Bounty     -1470.19231 -1676.50044 -1263.88418 0.0000000
0:Decorator-15:Bounty      348.07692   141.76879   554.38505 0.0000105
5:Decorator-15:Bounty    -1166.34615 -1372.65428  -960.03802 0.0000000
15:Decorator-15:Bounty   -1425.00000 -1631.30813 -1218.69187 0.0000000
5:Comfort-0:Comfort      -1494.23077 -1700.53890 -1287.92264 0.0000000
15:Comfort-0:Comfort     -2804.80769 -3011.11582 -2598.49956 0.0000000
0:Decorator-0:Comfort     -986.53846 -1192.84659  -780.23033 0.0000000
5:Decorator-0:Comfort    -2500.96154 -2707.26967 -2294.65341 0.0000000
15:Decorator-0:Comfort   -2759.61538 -2965.92351 -2553.30725 0.0000000
15:Comfort-5:Comfort     -1310.57692 -1516.88505 -1104.26879 0.0000000
0:Decorator-5:Comfort      507.69231   301.38418   714.00044 0.0000000
5:Decorator-5:Comfort    -1006.73077 -1213.03890  -800.42264 0.0000000
15:Decorator-5:Comfort   -1265.38462 -1471.69275 -1059.07649 0.0000000
0:Decorator-15:Comfort    1818.26923  1611.96110  2024.57736 0.0000000
5:Decorator-15:Comfort     303.84615    97.53802   510.15428 0.0002268
15:Decorator-15:Comfort     45.19231  -161.11582   251.50044 0.9989206
5:Decorator-0:Decorator  -1514.42308 -1720.73121 -1308.11495 0.0000000
15:Decorator-0:Decorator -1773.07692 -1979.38505 -1566.76879 0.0000000
15:Decorator-5:Decorator  -258.65385  -464.96198   -52.34572 0.0035990

plot(TukeyHSD(towelanova1,conf.level = .95))

# Compute Minimum Significant Difference w for interactions

# q(p,v) Critical studentized range
q<-qtukey(.95,nmeans=9,df=225)
q

[1] 4.429682

s<-sqrt(56398)
n<-26
w<-q*(s/sqrt(n))
w

[1] 206.3088

• We can see that most pairs are significantly different.

Step 5: Assumptions

Normality

We would like to evaluate the distribution of each treatment.

# NOTE: this function is qqPlot from the car package
qqPlot(toweldata$Strength, groups=toweldata$Brand)

qqPlot(toweldata$Strength, groups=toweldata$Water)

## EXTRA PLOTTING NOTES
# There are a few options to create plots for each treatment
# So far we have been using base R plotting.
# We can use a loop to make histograms for each level of one factor
par(mfcol=c(2, 2))
for (i in unique(toweldata$Brand)){
  hist(toweldata$Strength[toweldata$Brand==i], xlab="Strength", main=paste("Histogram of Brand", i))
}

unique(toweldata$Brand)

[1] "Decorator" "Comfort"   "Bounty"   

# We can use dev.off() to clear the plot formatting
# dev.off()

# Making plots by two factors is even more complicated in base R plotting
# The package ggplot2 with the function ggplot() is useful for 
# more customizable graphics.

towelplot<-ggplot(toweldata, aes(x=Strength))+ geom_histogram(bins=12, center=.5)+ facet_grid(Brand~Water)
towelplot

• The population distributions of the response do not appear to meet the normality assumption.

Equal Variance

Since Normality does not seem reasonably met, we will use Levene’s Test of equal variance.

leveneTest(Strength ~ Brand*factor(Water), data = toweldata,center=median)

boxplot(Strength ~ Brand+Water, data = toweldata)

• There is not evidence that the constant variance assumption is met. While these data meet the threshold of treatment sizes of “20”, it is recommended that we either transform the response variable (log, square root, etc) to better fit the assumptions. However, recall when we interpret the response, we cannot give as practical interpretations of the mean of the response. So, instead we may pursue a different “non-parametric” type of analysis.