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### Analysis of Variance and Multiple Comparisons

Comparing more than two means and figuring out which are different

Analysis of Variance (ANOVA)

- Despite the name, the procedures compares the means of two or more groups
- Null hypothesis is that the group means are all equal
- Widely used in experiments, it is less common in anthropology

ANOVA in Rcmdr

- Statistics | Means | One-way ANOVA
- Accept or change the model name
- Select a group (only factors are listed here)
- Select a response variable (only numeric variables are listed here)
- Check Pairwise comparison of means

> AnovaModel.1 <- aov(Area ~ Segment, data=Snodgrass)

> summary(AnovaModel.1)

Df Sum Sq Mean Sq F value Pr(>F)

Segment 2 432327 216164 51.817 1.344e-15 ***

Residuals 88 367107 4172

---

Signif. codes: 0 \'***\' 0.001 \'**\' 0.01 \'*\' 0.05 \'.\' 0.1 \' \' 1

> numSummary(Snodgrass$Area , groups=Snodgrass$Segment,

+ statistics=c("mean", "sd"))

mean sd n

1 317.3711 76.08797 38

2 166.7946 59.99526 28

3 192.7900 48.18188 25

Results

- Since the ANOVA statistic is less than our critical value (.05), we reject the null hypothesis that the mean Areas of Segments 1 = 2 = 3
- But we usually want to know more
- Since we did not make predictions in advance our comparisons are post hoc

Multiple Comparisons

- To find out which means are different from each other we have to compare the various combinations: 1 with 2, 1 with 3, and 2 with 3
- (we could also perform other comparisons such as 1 and 2 with 3 but they are rare in anthropology

More Kinds of Errors

- Our statistical tests have focused on setting the Type I error rate at .05 – the comparisonwise error rate
- But this error rate holds for a single test. If we do many tests, the chance that we will commit at least one Type 1 error will be higher – the experimentwise error rate

Calculating Errors

- If the probability of a Type I error is .05, the probability of not making a Type I error is (1 - .05) = .95
- The probability of not making a Type I error twice is .952 = .9025, three times - .953 = .8574, four times - .954 = .8145

Calculating Errors

- The probability of making at least one Type I error is
- Twice – (1 - .9025) = .0975
- Thrice – (1 - .8574) = .1426
- Four times – (1 - .8145) = .1855
- The probability of making at least one Type I error increases with each additional test

curve((1-(1-.05)^x), 1, 50, 50, yaxp=c(0, .9, 9),

xaxp=c(0, 50, 10), xlab="Number of Comparisons",

ylab="Type I Error Rate", las=1,

main="Experimentwise Error Rate")

curve((1-(1-.01)^x), 1, 50, 50, lty=2, add=TRUE)

text(30, .92, expression(p == 1-(1-.05)^x), pos=4)

text(30, .37, expression(p == 1-(1-.01)^x), pos=4)

abline(h=seq(.1, .9, .1), v=seq(0, 50, 5), lty=3, col="gray")

legend("topleft", c("Comparisonwise p = .05",

"Comparisonwise p = .01"), lty=c(1, 2), bg="white")

Multiple Comparisons

- Multiple Comparisons procedures take experimentwise error into account when comparing the group means
- There are a number of methods available, but we’ll stick with Tukey’s Honestly Significant Differences (aka Tukey’s range test)

Tukey’s HSD

- One of the few multiple comparison tests that can adjust for different sample sizes among the groups
- You requested this test in Rcmdr when you checked “Pairwise comparison of the means”

> .Pairs <- glht(AnovaModel.1, linfct = mcp(Segment = "Tukey"))

> summary(.Pairs) # pairwise tests

Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts

Fit: aov(formula = Area ~ Segment, data = Snodgrass)

Linear Hypotheses:

Estimate Std. Error t value Pr(>|t|)

2 - 1 == 0 -150.58 16.09 -9.361 <1e-04 ***

3 - 1 == 0 -124.58 16.63 -7.490 <1e-04 ***

3 - 2 == 0 26.00 17.77 1.463 0.313

---

Signif. codes: 0 \'***\' 0.001 \'**\' 0.01 \'*\' 0.05 \'.\' 0.1 \' \' 1

(Adjusted p values reported -- single-step method)

> confint(.Pairs) # confidence intervals

Simultaneous Confidence Intervals

Multiple Comparisons of Means: Tukey Contrasts

Fit: aov(formula = Area ~ Segment, data = Snodgrass)

Quantile = 2.383

95% family-wise confidence level

Linear Hypotheses:

Estimate lwrupr

2 - 1 == 0 -150.5764 -188.9093 -112.2435

3 - 1 == 0 -124.5811 -164.2161 -84.9460

3 - 2 == 0 25.9954 -16.3553 68.3460

NonParametric ANOVA

- The non-parametric alternative to ANOVA is the Kruskal-Wallis Rank Sum Test
- The null hypothesis is that the medians of the groups are equal
- If the test is significant, a multiple comparison method is available to identify which groups are different

Kruskal-Wallis in Rcmdr

- Statistics | Nonparametric tests | Kruskal-Wallis test
- Select a group (only factors are listed here)
- Select a response variable (only numeric variables are listed here)

Multiple Comparisons

- If there are significant differences the function kruskalmc() in package pgirmess will tell you what groups are different

> kruskal.test(Area ~ Segment, data=Snodgrass)

Kruskal-Wallis rank sum test

data: Area by Segment

Kruskal-Wallis chi-squared = 50.4427, df = 2, p-value = 1.113e-11

library(pgirmess)

> kruskalmc(Area ~ Segment, data=Snodgrass)

Multiple comparison test after Kruskal-Wallis

p.value: 0.05

Comparisons

obs.dif critical.dif difference

1-2 43.125940 15.74873 TRUE

1-3 35.227368 16.28369 TRUE

2-3 7.898571 17.39936 FALSE

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