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Analysis of Variance and Multiple Comparisons Analysis of Variance and Multiple Comparisons

Analysis of Variance and Multiple Comparisons - PowerPoint Presentation

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Analysis of Variance and Multiple Comparisons - PPT Presentation

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 ID: 243718

comparisons error multiple test error comparisons test multiple means segment type area kruskal snodgrass wallis anova data groups rate

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Slide1

Analysis of Variance and Multiple Comparisons

Comparing more than two means and figuring out which are differentSlide2

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 anthropologySlide3

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 meansSlide4

> 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 25Slide5

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 hocSlide6

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 anthropologySlide7

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 rateSlide8

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 .95

2

= .9025, three times - .95

3

= .8574, four times - .95

4

= .8145Slide9

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 testSlide10
Slide11

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")Slide12

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)Slide13

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”Slide14

> .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)Slide15

>

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

lwr

upr

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 68.3460Slide16
Slide17

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 differentSlide18

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)Slide19

Multiple Comparisons

If there are significant differences the function

kruskalmc

()

in package

pgirmess

will tell you what groups are differentSlide20

>

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