Tukey’s Method Continued

Here are the sampling results of a product from four suppliers, A, B, C and D (Data courtesy: https://statisticsbyjim.com/).

ABCD
40 37.9 36 38
36.9 26.2 39.4 40.8
33.4 24.936.3 45.9
42.3 30.3 29.5 40.4
39.1 32.6 34.9 39.9
34.737.539.841.4

Hypotheses

N0 – All means are equal
NA – Not all means are equal

Input the data

PO_data <- read.csv("./Anova_Tukey.csv")
as_tibble(PO_data)

Leads to the output (first ten entries)

Material Strength
<chr>     <dbl>
B	  37.9			
C	  36.0			
D	  38.0			
A	  40.0			
A	  36.9			
C	  39.4			
A	  33.4			
B	  26.2			
B	  24.9			
B	  30.3

Plot the data

par(bg = "antiquewhite")
colors = c("red","blue","green", "yellow")
boxplot(PO_data$Strength ~ factor(PO_data$Material), xlab = "Supplier", ylab = "Material Data", col = colors)

F-test for ANOVA

str.aov <- aov(Strength ~ factor(Material), data = PO_data)
summary(str.aov)

Output:

                 Df Sum Sq Mean Sq F value Pr(>F)   
factor(Material)  3  281.7    93.9   6.018 0.0043 **
Residuals        20  312.1    15.6                  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Reject null hypothesis

We at least reject the null hypothesis because the p-value < 0.05 (the chosen significance level). The F-value is 6.018. Another way of coming the conclusions is to find out the critical F value for the degrees of freedoms, df1 = 3 and df2 = 20.

qf(0.05, 3,20, lower.tail=FALSE)
pf(6.018, 3, 20, lower.tail = FALSE)

Lead to

3.098391      # F-critical
0.004296141   # p-value for F = 6.018

Tukey’s test for multiple comparisons of means

TukeyHSD(str.aov)
Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = Strength ~ factor(Material), data = PO_data)

$`factor(Material)`
         diff        lwr        upr     p adj
B-A -6.166667 -12.549922  0.2165887 0.0606073
C-A -1.750000  -8.133255  4.6332553 0.8681473
D-A  3.333333  -3.049922  9.7165887 0.4778932
C-B  4.416667  -1.966589 10.7999220 0.2449843
D-B  9.500000   3.116745 15.8832553 0.0024804
D-C  5.083333  -1.299922 11.4665887 0.1495298

Interpreting pair-wise differences

You can see that the D-B difference, 9.5, is statistically significant at an adjusted p-value of 0.0022. And, as expected, the 95% confidence interval for D-B doesn’t include 0 (no difference between D and B).

By the way, the message, the difference between blue box and yellow, was already apparent had you paid attention to the box plots we made in the beginning.

Reference

Hypothesis Testing: An Intuitive Guide: Jim Frost

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