Biol 500: basic statistics Goals: 1) understand basics of experimental design - controls - replication 2) understand how to report quantitative data 3) be able to interpret the reporting of statistical results in a journal article

Replication: allows you to determine if the difference between treatments or groups of samples is greater than the variation within a treatment or group Is there a difference in how effective the 3 drugs are in curing headaches?

Replication: allows you to determine if the difference between treatments or groups of samples is greater than the variation within a treatment or group Is there a difference in how effective the 3 drugs are in curing headaches? Generally, overlapping error bars indicate no significant difference between the mean values that are being graphed Bars don’t overlap = probably different ? no yes

Controls: From these data, could you tell if the least effective drug has any effect at all?

Controls: From these data, could you tell if the least effective drug has any effect at all? Including a control that is the same in all respects except the key variable you are manipulating is key to interpreting your results

Controls: Procedural controls allow you to diagnose problems in your experiment, samples or technique When we amplify DNA from unknown samples by PCR, we include a positive control (a DNA sample that always works) and a negative control (all the PCR reagents, but no DNA) This allows us to interpret the results of our gels, and to troubleshoot any problems

Do squirrels bury acorns? My experiment: I remove all the squirrels from 3 clumps of trees in one park, but leave the squirrels 3 control clumps of trees in another park on the other side of town Park A Park B

Pseudoreplication In this example the unit of replication is the park, not the clump of trees – I have no actual replication Park A Park B Any difference that I measure could be due to differences among the two parks, and not due to my squirrel-removal treatment

Avoiding pseudoreplication Correct design would be to have squirrel-removal and control areas in each of several replicate parks This lets you assess differences between treatment and control areas, while simultaneously measuring variation among parks Park C Park A Park B

n = 10 Did these two classes do differently on my 418 midterm?

n = 20

n = 44

n = 44 The statistical approach is to ask if the means of these X = 133.9 ± 29.7 SD range: 59 - 183 X = 126.3 ± 38.8 SD range: 42 - 188 The statistical approach is to ask if the means of these two populations are significantly different

n = 44 the standard deviation (SD) is what you should report if you X = 133.9 ± 29.7 SD range: 59 - 183 X = 126.3 ± 38.8 SD range: 42 - 188 the standard deviation (SD) is what you should report if you are actually interested in the variation – ie, for purposes of deciding where to draw the line between grades

√ X = 133.9 ± 29.7 SD or ± 4.3 SE range: 59 - 183 n = 44 the standard error (SE, or SEM) is SD √ n sample size

X = 133.9 ± 29.7 SD or ± 4.3 SE range: 59 - 183 n = 44 X = 126.3 ± 38.8 SD or ± 5.8 SE range: 42 - 188 the standard error is what you report when you want to compare the means of different treatments or samples

unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 X = 133.9 ± 29.7 SD X = 126.3 ± 38.8 SD unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 a t-test compares 2 populations by calculating a test statistic called t and determining the probability (P, or p) of getting that value of t, with that sample size, by chance alone

unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 X = 133.9 ± 29.7 SD X = 126.3 ± 38.8 SD unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 paired would be, you compare the % scores on midterm versus final for each student; most tests are unpaired

unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 X = 133.9 ± 29.7 SD X = 126.3 ± 38.8 SD unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 one-tailed if you have some reason to think, in advance, that the 2009 scores will only be higher (or lower) than 2007 - cuts your P-value in half, but you need a reason to do this!

unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 X = 133.9 ± 29.7 SD X = 126.3 ± 38.8 SD unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 power of your test will depend on your degrees of freedom, which is (sample size) – (number of groups) - in this case: (44 + 44 students) – (2 groups) = 88 -2 = 86

unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 X = 133.9 ± 29.7 SD X = 126.3 ± 38.8 SD unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23 P values below 0.05 are accepted as significant, meaning there is less than a 5% chance of getting a test statistic this large if the groups are not really any different

3 or more samples can be compared using a df subscripted under F ratio F2,129 = 7.12 P <0.001 overall P for 3-way comparison of means n = 44 n = 44 n = 44 3 or more samples can be compared using a one-way Analysis of Variance, or ANOVA instead of calculating a t statistic, ANOVA calculates an F-ratio, which compares variation within groups (error bars) to the differences in mean values among groups 2 degrees of freedom: 1st = (# of groups – 1) 2nd = (total sample size) – (# of groups)

If your overall P-value is significant, you can then do a post-hoc Scheffe: P = 0.002 Scheffe: P = 0.050 n = 44 n = 44 n = 44 If your overall P-value is significant, you can then do a post-hoc (“after the fact”) test to work out which specific means are different from each other Bonferroni - not too conservative; may see differences that aren’t real Scheffe - very conservative; if it sees a difference, there really is one Dunnett - compares each mean to a control; most powerful

2-way ANOVA tests for interactions among 2 or more factors factors: aspirin, yes/no tylenol, yes/no

2-way ANOVA tests for interactions among 2 or more factors when the response to two treatments combined is not what you would expect from adding their individual effects, this is an interaction interactions are usually the most biologically interesting result!

2-way ANOVA tests for interactions among 2 or more factors A B C D NOT appropriate to do a 1-way ANOVA on these data, because that requires that each treatment be independent of the other treatments - since 2 treatments involve aspirin, they are not independent - also, you miss the interaction, which is the important result

Correlation analysis is appropriate when you think 2 variables are related, but not in a cause-and-effect way - arm length and leg length are related, but longer arms do not cause you to have longer legs; both are due to your height Regression analysis is when you believe a change in one predictor variable (what you manipulate) causes a change in the response variable (the thing you measure) - adding more water makes plants grow taller

Output of a regression analysis includes: 1) ANOVA table tells you if your model explains a significant amount of the variation in the response

Output of a regression analysis includes: 1) ANOVA table 2) equation of the best-fit line summarizes the relationship between predictor and response

Output of a regression analysis includes: 1) ANOVA table 2) equation of the best-fit line 3) table testing the effect of each predictor in multiple regression, you can test many possible predictors that might matter, and see which significantly affect the response variable

Output of a regression analysis includes: 1) ANOVA table 2) equation of the best-fit line 3) table testing the effect of each predictor 4) r2 r2 is the % of variation in the response that is due a change in the predictor

More scatter = lower r2 You can have a low r2, but still have a significant slope

ANOVA and regression are both types of linear models, which test the same basic equation: response variable = model + error variance in the response that is not explained by the model thing you measure predictors, and coefficients that tell you how they affect the response this is what a simple linear regression model looks like

Does predictor X affect response? test is to set the coefficient = 0, which drops out the predictor, and see if the model (now just the residual error term) is really any worse

Parametric versus non-parametric tests All of the tests we have discussed are parametric tests - they use the numerical values of your actual data - however, they also have built-in assumptions that your data, and the residual errors, fit a normal distribution (bell curve)

Parametric versus non-parametric tests % scores If your data do not fit a normal distribution, you can transform the raw numbers to make them more “normal” – put the data through a mathematical function arcsine(square-root(%)) is a standard transformation for %’s which stop at 100%, and are often not normally distributed

Parametric versus non-parametric tests Alternatively, there are non-parametric versions of most common statistical tests that use ranked values instead of the raw data - are typically more conservative: if they see a difference, it is real - make no assumptions about the shape of the distribution raw ranked (high to low) 3 5 2 6 6 3 4 4 9 2 1 7 12 1