STAT 216 T-test Assumptions and Limitations

T-test Assumptions

1. random sampling

2. normal distribution - CLT applies, but heavy tailed data bad!

3. independent observations - think cluster and serial effect

4. homogeneity of variance - particularly bad when sample sizes differ

Shapiro-Wilk Normality Test

• null hypothesis: data comes from normally distributed population

• W-test statistic + p-value to interpret results

• note: test is sensitive to sample size - large samples can produce type II error

Testing Homogeneity of Variance

1. Hartley’s test: largest sample variance/smallest sample variance BUT requires equal sample size, normality, NON ROBUST

2. Levene’s test: MSbetween/MSwithin, MODERATELY ROBUST

3. Brown-Forsythe test: uses median, resistant to outliers, MOST ROBUST 

Durbin-Watson Test of Independence

• tests non-independent observations (autocorrelation)

• DW<2 = Positive autocorrelation (residuals have same sign)

• DW>2 = Negative autocorrelation (residuals alternate sign)

Outliers

• z-score rule: 2SD = 5% are outliers, 3SD = <1% are outliers NON ROBUST

• boxplot rule: ROBUST

• how outliers effect inference with t:

  • disturb point estimates

  • contribute to skew/heavy tails

  • can affect nonrobust tests of assumptions

T-inferences are inaccurate when…

1. non-normality is heavy-tailed

2. data come from nonrandom samples

3. heterogeneous variances + small sample sizes

4. nonindependence (serial, cluster, protocol)