From the Gilly lab, Oxygen Consumption of Jumbo Squid ignore the grouping and repeated measurements and just take 28 observations H L M 10 8 10 in the 3 O2 groups. respiration rates (milligrams oxygenconsumed per hr) normalized by body mass -- so final units are mg/kg/hr. There are 3 conditions of oxygen concentration where the respiration rates are measured: High (220-150 micromolar), Medium (110-35) and Low (13-7.5). respiration rate is labeled as mgperkgperhr because Low O2 has such small outcomes there is a big difference in spread > tapply(mgperkgperhr, O2, mean) > tapply(mgperkgperhr, O2, sd) H L M H L M 161.53421 17.46748 115.14973 54.19495 13.60296 44.56000 > tapply(mgperkgperhr, O2, quantile) $H 0% 25% 50% 75% 100% 63.91072 128.01847 192.66966 200.57419 211.69167 $L 8.467667 9.071240 11.362161 17.919569 44.141836 $M 55.74547 80.37983 110.71199 137.64082 185.0790 square root transform (reflex when sd increases with mean) does stabilize sd's > tapply(sqrt(mgperkgperhr), O2, sd) H L M 2.361193 1.434322 2.085932 but since sample sizes are close not that much diff in testing overall at least. > squidaov= aov(mgperkgperhr ~ O2) #so do some anovas > summary(squidaov) Df Sum Sq Mean Sq F value Pr(>F) O2 2 94247 47123 25.836 8.248e-07 *** Residuals 25 45599 1824 so diff between 3 levels highly signif even when procedure assumes equal vars Compare with a version that does not assume equal vars (Welch is buzzword) > oneway.test(mgperkgperhr~ O2) One-way analysis of means (not assuming equal variances) data: mgperkgperhr and O2 F = 47.7383, num df = 2.000, denom df = 13.942, p-value = 5.79e-07 details differ but same basic result, big effect for O2 across 3 levels But more interesting question is pairwise comparisons--unless you drop down to overall confidence level .9 , Med vs High diff is not distinguishable, Confidence interval includes 0, whether or not you use sqrt transform to stabilize variances > TukeyHSD(squidaov) | same if you do sqrt transform Tukey multiple comparisons of means | > squidaovsqrt= aov(sqrt(mgperkgperhr) ~ O2) 95% family-wise confidence level | > summary(squidaovsqrt) | Df Sum Sq Mean Sq F value Pr(>F) Fit: aov(formula = mgperkgperhr ~ O2) | O2 2 346.78 173.39 41.786 1.066e-08 *** | Residuals 25 103.74 4.15 $O2 | --- diff lwr upr | > TukeyHSD(squidaovsqrt) L-H -144.06673 -194.52649 -93.606966 | Tukey multiple comparisons of means M-H -46.38449 -93.95840 1.189435 | 95% family-wise confidence level M-L 97.68224 47.22248 148.142005 | Fit: aov(formula = sqrt(mgperkgperhr) ~ O2) | $O2 > TukeyHSD(squidaov, c=.9) | diff lwr upr Tukey multiple comparisons of means | L-H -8.552453 -10.959223 -6.1456839 90% family-wise confidence level | M-H -1.963916 -4.233040 0.3052086 Fit: aov(formula = mgperkgperhr ~ O2) | M-L 6.588538 4.181768 8.9953074 $O2 |------------------------------------------------- diff lwr upr |unless you drop down to overall confidence L-H -144.06673 -187.62963 -100.503822 |level .9 , Med vs High diff is not M-H -46.38449 -87.45599 -5.312983 |distinguishable, Confidence interval includes 0, M-L 97.68224 54.11934 141.245149 |whether or not you use sqrt transform |to stabilize variances --