Variances, Fallacies, and Exams - Oh my!!!

• Reading Assignment for Friday (NO QUIZ)
  • Whitlock & Schluter, Chapter 13: Handling violations of assumptions

Question: Do populations differ in the variability of measurements?

Remember, it isn't always about inferring central tendency!

There are two main tests:

• \( F \)-test (Warning: Highly sensitive to departures from normality assumption)
• Levene's test (More robust to departures from normality, but at a cost - loss of power!)

Example 12.4

The brook trout is a species native to eastern North America that has been introduced into streams in the West for sport fishing. Biologists followed the survivorship of a native species, chinook salmon, in a series of 12 streams that either had brook trout introduced or did not (Levin et al. 2002). Their goal was to determine whether the presence of brook trout effected the survivorship of the salmon. In each stream, they released a number of tagged juvenile chinook and then recorded whether or not each chinook survived over one year.

Load data and sneak-a-peek:

'data.frame':   12 obs. of  4 variables:
 $ troutTreatment    : Factor w/ 2 levels "absent","present": 2 1 2 2 1 2 1 2 1 1 ...
 $ nReleased         : int  820 467 960 700 959 545 1029 769 27 998 ...
 $ nSurvivors        : int  166 180 136 153 178 103 326 173 7 120 ...
 $ proportionSurvived: num  0.202 0.385 0.142 0.219 0.186 0.189 0.317 0.225 0.259 0.12 ...

Compute variances in both groups:

(vari <- tapply(chinook$proportionSurvived, 
                chinook$troutTreatment, 
                var))

      absent      present 
0.0107413667 0.0008829667 

var.test(proportionSurvived ~ troutTreatment, 
         data = chinook)

    F test to compare two variances

data:  proportionSurvived by troutTreatment
F = 12.165, num df = 5, denom df = 5, p-value = 0.01589
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
  1.702272 86.936360
sample estimates:
ratio of variances 
          12.16509 

library(car)
leveneTest(chinook$proportionSurvived, 
           group = chinook$troutTreatment, 
           center = mean)

Levene's Test for Homogeneity of Variance (center = mean)
      Df F value   Pr(>F)   
group  1  10.315 0.009306 **
      10                    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

How to compare between two groups with only confidence intervals?

Example 12.5: Mommy's baby, Daddy's maybe

Question: Do babies look more like their fathers or their mothers?

Example 12.5: Mommy's baby, Daddy's maybe

Question: Do babies look more like their fathers or their mothers?

Christenfeld and Hill (1995) predicted that babies more resemble their fathers, due to the hypothesis that this resemblance affords an evolutionary advantage of increased paternal care. They tested this by obtaining pictures of a series of babies and their mothers and fathers. Particpants shown picture of child, and either three possible mothers or three possible fathers (one is correct).

Conclusion: Authors concluded that since fathers turned up statistically significant and mothers did not, that babies more resembled their fathers than their mothers.

Discuss: What’s the mistake here?

Mistake: Misinterpretation of statistical significance

Fallacy: If one test in Group 1 shows with statistical significance that \( \mu_{1} > \mu_{0} \), and the same test in Group 2 does not show \( \mu_{2} > \mu_{0} \), then this shows with statistical significance that \( \mu_{1} > \mu_{2} \).

Fallacy: If one test in Group 1 shows with statistical significance that \( \mu_{1} > \mu_{0} \), and the same test in Group 2 does not show \( \mu_{2} > \mu_{0} \), then this shows with statistical significance that \( \mu_{1} > \mu_{2} \).

Fallacy: If \( \bar{Y}_{1} > \bar{Y}_{2} \), then \( \mu_{1} > \mu_{2} \).

Mistake: Relying on point estimates rather than interval estimates

Conclusion: Comparisons between two groups should always be made directly using the appropriate statistical test, not indirectly by comparing both to the same null hypothesized value.