Comparing Multiple Groups: Analysis of Variance ANOVA (1-way) Scientific Practice Comparing Multiple Groups: Analysis of Variance ANOVA (1-way)

Where We Are/Where We Are Going We have looked at the 1-sample t-test and the 2-sample t-test 1-sample compares sample mean vs expected mean paired t-test is a special case of ‘Before minus After’ 2-sample compares two samples against each other eg Males vs Females But what if we want to compare lots of different situations? repeatedly applying a t-test… takes time increase the chance of anomalous result does not consider all data together

One-Way ANOVA One-Way refers to the type of experiment being considered, not the number of conditions eg we can compare effects of 10 different BP drugs compare growth rates of 7 different bacteria Like t-tests, we want to know if means differ but ANOVA examines variability of underlying data Assumptions… data are normally distributed samples are independent variances of each group are similar (not that strict) ANOVA partitions total variability into… between-groups and within-groups

One-Way ANOVA Step 1: Formulate the Null Hypothesis groups do not vary in terms of their means Step 2: Generate the Test Statistic, F F = MSbetween/MSwithin where MS = s2 note if Null Hypo is true, then the random things determining MSwithin soley determine MSbetween ie F = 1 but the more the total variability can be assigned to between-group and the less to within-groups… …the bigger F becomes ie a bigger F implies differences between groups that could be significant

One-Way ANOVA Step 3: Work out probability the F statistic has a distribution, the F distribution note one tail origin of 0 peak around 1 no negatives the usual logic applies… a ‘critical value’ for F can be looked up in a table if our F is greater, then p < 0.05 (or other level) critical F depends on DoF of between-groups and within-groups between-groups DoF = #groups – 1 within-groups DoF = total readings – #groups

One-Way ANOVA

One-Way ANOVA

One-Way ANOVA Step 4: Interpret the probability ‘Issues’ with ANOVA if p < 0.05, then the Null Hypo is rejected… …in favour of the Alt Hypo, that there is at least one significantly different mean in the groups we are investigating ‘Issues’ with ANOVA My ANOVA tells me something is going on, but it doesn’t tell me what! how do I find out which group(s) is/are sig different? Eg which of my 10 BP drugs is having an effect? Terminology eg MS

One-Way ANOVA: Terminology Between groups aka… factor (Minitab), treatment Within groups aka… error (Minitab), residual Example using Minitab… 4 cages of rats kept on a control diet are there differences in weights between cages?

One-Way ANOVA: Terminology Minitab output (part)… Source DF SS MS F P Factor 3 27234 9078 2.27 0.119 Error 16 63954 3997 Total 19 91188 Factor (between), Error (within) DF = degrees of freedom SS = sum of the squares each weight is a distance (+/-) from the mean and each group mean is (+/-) from overall mean square each ‘distance’ to remove the sign sum each together to get indication of variability

One-Way ANOVA: Sum of Squares

One-Way ANOVA: Terminology Minitab output (part)… Source DF SS MS F P Factor 3 27234 9078 2.27 0.119 Error 16 63954 3997 Total 19 91188 MS = Mean Square simple… MS = SS / DF less simple… MS is the variance of each factor variance = sum of squared differences / N where N is DoF, not number of observations F = MSFactor / MSError = 9078/3997 = 2.27

One-Way ANOVA: What changed? Say, unknown to us, a cage was unlocked and rats kept sneaking out to eat extra food… Source DF SS MS F P Factor 3 193834 64611 16.16 0.000 Error 16 63954 3997 Total 19 257788 Critical F (3, 16 DoF; 0.05 level) = 3.24 note Minitab calculates actual p (<0.0005% ) v low chance variation between groups by chance reject Null Hypo; but which one(s) are different?

One-Way ANOVA: What changed? When ANOVA flags an effect, working out what has changed is called post hoc testing comparing all groups with each other – Tukey test comparing groups to a control – Dunnett test Tukey in Minitab for a ‘family’ of comparisons, Minitab ‘ups’ the 95% confidence level for individual comparisons so the ‘family’ level is 95% Grouping Information Using Tukey Method N Mean Grouping Cage 2 5 523.20 A Cage 4 5 371.20 B Cage 1 5 290.40 B Cage 3 5 274.80 B Means that do not share a letter are significantly different.

One-Way ANOVA: What changed? Dunnett test creates CIs for differences between mean of each ‘expt’ group and mean of control group if a CI contains 0, then no significant difference as before, multiple comparisons means that individual confidence levels increased so ‘family’ confidence level appropriate (95%) It’s stupid, but say our unlocked cage was the control eg looking at effect of locking cages on weight of rats

One-Way ANOVA: What changed? Level N Mean Grouping Cage 2 (control) 5 523.20 A Cage 4 5 371.20 Cage 1 5 290.40 Cage 3 5 274.80 Means not labeled A are sig different to control level mean Family error rate = 0.05 Individual error rate = 0.0196 Intervals for treatment mean minus control mean Level Lower Center Upper Cage 1 -336.46 -232.80 -129.14 Cage 3 -352.06 -248.40 -144.74 Cage 4 -255.66 -152.00 -48.34 Tells us the ‘expt’ cages are different to the control cage CI does not include 0 difference

One-Way ANOVA: Is that it? 2-sample t-test and 1-way ANOVA differ as 1-way ANOVA can deal with >2 groups but both rely on sets of data being… normally distributed independent of each other ie unpaired data Is there an equivalent ANOVA to ‘paired t-test’? eg same group of subjects given a series of drugs two solutions the repeated measures ANOVA but not trivial to do! Leave until later? randomised block design

Summary 1-way ANOVA allows many groups to be compared in terms of one factor (treatment) Partitions total variability between-groups and within-groups Test Statistic, F = MSbetween/MSwithin big F implies significant differences Post hoc testing needed if ANOVA significant Tukey test for testing all groups against each other Dunnett test for comparing ‘expt’ groups v a control 1-way ANOVA an ‘extension’ of unpaired t-test Repeated Measured/Randomised Block ANOVA for paired data