Topics

• Compare groups using analysis of variance

Aldrin in the Wolf River

• The Wolf River in Tennessee flows past an abandoned site once used by the pesticide industry for dumping wastes, including chlordane (pesticide), aldrin, and dieldrin (both insecticides).

• These highly toxic organic compounds can cause various cancers and birth defects.

Aldrin in the Wolf River

## # A tibble: 30 x 2
##    aldrin depth 
##     <dbl> <chr> 
##  1    3.8 bottom
##  2    4.8 bottom
##  3    4.9 bottom
##  4    5.3 bottom
##  5    5.4 bottom
##  6    5.7 bottom
##  7    6.3 bottom
##  8    7.3 bottom
##  9    8.1 bottom
## 10    8.8 bottom
## # … with 20 more rows

Aldrin in the Wolf River

• The standard methods to test whether these substances are present in a river is to take samples at six-tenths depth.

• These compounds are denser than water and their molecules tend to stick to particles of sediment, they are more likely to be found in higher concentrations near the bottom than near mid-depth.

Aldrin by depth

Notation

• K is number of mutually exclusive groups. We index the groups as i=1,…,K.
  

• ni is number of observations in group i
  

• n=n1+n2+⋯+nK is the total number of observations in the data
  

• yij is the jth observation in group i, for all i,j
  

• μi is the population mean for group i, for i=1,…,K

Using ANOVA to compare means

• Question of interest Is the mean value of the response y the same for all groups, or is there at least one group with a significantly different mean value?

• To answer this question, we will test the following hypotheses:

What's happening...

• If the sample means are "far apart", " there is evidence against H0

• We will calculate a test statistic to quantify "far apart" in the context of the data

Analysis of Variance (ANOVA)

Main Idea: Decompose the total variation in the data into the variation between groups (model) and the variation within each group (residuals)

K∑i=1ni∑j=1(yij−ˉy)2=K∑i=1ni(ˉyi−ˉy)2+K∑i=1ni∑j=1(yij−ˉyi)2

• If the variation between groups is significantly greater than the variation within each group, then there is evidence against the null hypothesis.

ANOVA table

Total variation

Total variation: variation between and within groups

SSTotal=16.961+37.329=54.290

DFTotal=2+37=29

s2y=SSTotalDFTotal=54.29029=1.872

Between variation

Between variation: variation in the group means

SSBetween=16.961

DFBetween=2

MSBetween=SSBetweenDFBetween=15.9612=8.480

Within variation

Within variation: variation within each group

SSWithin=37.329

DFWithin=27

MSWithin=SSWithinDFWithin=37.32927=1.383

Using ANOVA table to test difference in means

Using ANOVA table to test difference in means

Test statistic: Ratio of between group and within group variation

F=MSBetweenMSWithin=8.4801.383=6.134

Calculate p-value

Calculate the p-value using an F distribution with K−1 and n−K degrees of freedom

Using ANOVA table to test difference in means

P-value: Probability of observing a test statistic at least as extreme as F Stat given the group means are equal

The p-value is very small (≈0), so we reject H0. The data provide sufficient evidence that at least one depth level has a mean aldrin concentration that differs from the others.

Assumptions for ANOVA

1️⃣ Normality: yij∼N(μi,σ2)

2️⃣ Constant variance: The population distribution for each group has a common variance, σ2

3️⃣ Independence: The observations are independent from each other

• This applies to observations within and between groups

Checking Normality

✅ No major skewness or outliers.

Checking Normality

✅ Points fall relatively along the diagonal line.

Checking constant variance

## # A tibble: 3 x 4
##   depth        n  mean    sd
## * <chr>    <int> <dbl> <dbl>
## 1 bottom      10  6.04 1.58 
## 2 middepth    10  5.05 1.10 
## 3 surface     10  4.2  0.660

Checking independence

✅ Based on what we know about the study, we have no reason to believe that the aldrin concentrations are not independent of each other.

Robustness to Assumptions

• Normality: yij∼N(μi,σ2)

  • ANOVA relatively robust to departures from Normality.
  • Concern when there are strongly skewed distributions with different sample sizes (especially if sample sizes are small, < 10 in each group)

• Independence: There is independence within and across groups

  • If this doesn't hold, should use methods that account for correlated errors

Robustness to Assumptions

• Constant variance: The population distribution for each group has a common variance, σ2
  • Critical assumption, since the pooled (combined) variance is important for ANOVA
  • General rule: Satisfied if SDmax/SDmin≤2. OK if this is somewhat >2 when sample sizes are small.

Recap

• Used ANOVA to compare means across groups

Acknowledgements

• Analysis example and map image from OpenIntro Statistics