There is a type of experimental design that is referred to as a repeated measures design, because observations are made on an experimental unit prior to treatment, and again after treatment. The advantage of this is that it eliminates the variation and potential for bias that is present when assigning different experimental units for treatment or to serve as a control. It accomplishes this reduction by examining the differences between the pre- and post-treatment observations. The example that we are going to use is the effect of caffeine on the heart rate of the cladoceran Daphnia magna. Heart rates for 10 individual D. magna in filtered pond water were observed (using a dissecting scope), and then those 10 individuals were exposed to a concentration of 80 mg/L of caffeine in filtered pond water for 30 minutes, and the measurements of heart rate were repeated. Clearly (I hope), an individual D. magna with an abnormally high or low normal heart rate will add variation to the data if two separate groups of individuals were used, but if we perform the analysis only on the change in heart rate, that variation no longer influences the analysis.
If the data that we collect comes from the same experimental unit pre- and post-treatment, we can refer to these samples as paired samples, and analyze them using a paired-sample t-test. The premise of the paired-sample t-test is simple: if there is no treatment effect, then the mean difference between the pre- and post-treatment observations should be zero. Thus, a paired-sample t-test is merely a single-sample t-test, comparing the mean of the differences against a population mean (μ) of zero.
If we calculate the difference (d) between each paired observation as:
Then we can test the null hypothesis:
By calculating our single sample ts-value as:
As before, the standard deviation in the denominator is the standard deviation of the means of the differences, and so you must calculate the standard error from the sample of differences.
Let's apply this to the Daphnia experiment. There are 2 sets of data in the Excel worksheet labelled "Daphnia", and we will focus first on the exposed data in columns A and B.
Question 10: Based on a paired-sample t-test, do the data suggest that the addition of caffeine significantly increased the heart rate of the individuals of D. magna?
While a repeated measures design does remove nuisance variation from the data set, the design used for the preceding question has some holes in it. While the same individuals were exposed to caffeine, an overly anal-retentive (does that hyphen belong there?) experimental designer will be quick to point out that there is no way of knowing whether the conditions other than the addition of caffeine remained unchanged from the first to the second trial. If nothing else, the individuals were older in the second trial than they were in the first trial. Just to be safe, a control group of Daphnia should be set up and tested in the same way (2 trials), but without the addition of caffeine. That is the procedure that produced the second set of data in columns D and E. Unfortunately, this gives us 2 sets of mean differences to work with, such that the single-sample t-test no longer is applicable. The good news is that according to the Shapiro-Wilk test, both sets of differences are normally distributed (control: W = 0.899; p = 0.211; caffeine: W = 0.957; p = 0.750). The bad news is that it is up to you to determine how best to make this statistical comparison. Good luck, and God bless...
Question 11: Was the change in heart rate for the caffeine-treated Daphnia significantly greater than that of the control Daphnia?
Welcome to the exhilarating world of inferential statistics! Disclaimer: no bacteria or animals were harmed, inconvenienced, or even observed in the production of these data sets. While the means and estimates of the error are based on real scientific endeavors, the data were generated by models (the R programs can be viewed HERE) in order to protect the innocent.
Save your Word document with the answers (did you remember to justify your choice of one- or two-tailed probabilities?) and the Excel workbook showing your work as: yourlastnameex7, and submit them to me via Blackboard.
Week 7 Objectives
Know when to apply, and how to conduct a single-sample t-test.
Know how to compare 2 sample means using a 2-sample t-test.
Know how to report the results of a test of significance.
Know the assumptions of the single and 2-sample t-test, know how to test those assumptions, and know the nonparametric alternative(s).
Understand the concept of a repeated-measures design, and know when and how to apply a paired-sample t-test.
Send comments, suggestions, and corrections to: Derek Zelmer