1. Statistics for Everyone Workshop Fall 2010 Part 3 Comparing 2 Conditions With a T Test Workshop by Linda Henkel and Laura McSweeney of Fairfield University Funded by the Core Integration Initiative and the Center for Academic Excellence at Fairfield University

2. Statistics as a Tool in Scientific Research: Comparing 2 Conditions With a T Test Teaching Tips The research will excite them, but the statistics part might frighten them (“Oh, I’m not good at that…”) so ALWAYS emphasize statistics as a tool for researchers to use to answer research questions Ask exciting and interesting research questions so that they want to find out the answer Be sure to cover the science basics (e.g., terminology); If they don’t hear it from you, where do they hear it? Don’t assume they already know

3. Statistics as a Tool in Scientific Research: Comparing 2 Conditions With a T Test Types of Research Questions Descriptive (What does X look like?) Correlational (Is there an association between X and Y? As X increases, what does Y do?) Experimental (Do changes in X cause changes in Y?)

4. Statistics as a Tool in Scientific Research  Start with the science and use statistics as a tool to answer the research question Get your students to formulate a research question: How often does this happen? Did all plants/people/chemicals act the same? What happens when I add more sunlight, give more praise, pour in more water?

5. Types of Statistical Procedures Descriptive: Organize and summarize data Inferential: Draw inferences about the relations between variables; use samples to generalize to the population

6. Hypothesis Testing in Experiments Independent variable (IV) (manipulated): Has different levels or conditions Presence vs. absence (Drug, Placebo) Amount (5mg, 10 mg, 20 mg) Type (Drug A, Drug B, Drug C) Quasi-Independent variable (not experimentally controlled) e.g., Gender Dependent variable (DV) (measured variable): Number of white blood cells, temperature, heart rate Do changes in the levels/conditions of the IV cause changes in the DV? If graphing, put IV on x-axis and DV on y-axis

7. Hypothesis Testing in Experiments Statistical test: the t test Used for: Comparing average score in one condition to average score in another condition Use when: The IV is categorical (with 2 levels) and the DV is numerical (interval or ratio scale), e.g., weight as a function of gender, # of white blood cells as a function of organism, mpg as a function of foreign/domestic

8. Types of T Tests: One Sample T Use when you have one sample and you want to compare it to a known population Participants get Drug B. Clinical trials with Drug A have already shown how much it reduced pain. Does Drug B provide even better pain relief, on average? Plants are exposed to 3 hrs artificial light. You already know how many blooms, on average, plants will make when they have no artificial light. Does the artificial light cause more blooms on average than in the population at large? Hybrid cars are driven on a test course to determine their mpg. You already know the average mpg gas- powered cars get on this course. Do the hybrids get better or worse mpg on average than the population at large?

9. Types of T Tests: Independent Samples T Use when you have a between-subjects design -- comparing if there is a difference between two separate (independent) groups Some people get the drug, and other people get the placebo. Which group on average has less pain? Some plants are exposed to 0 hrs of artificial light, and some are exposed to 3 hours. Which group has more blooms on average? Some cars are hybrid and some are gas- powered. Which type of car gets better fuel mileage on average?

10. Types of T Tests: Paired Samples T Use when you have a within-subjects design – each subject experiences all levels/conditions of the IV; observations are paired/dependent/matched Each person gets both the drug and the placebo at different times. Which relieves pain better on average? Each plant is exposed to 0 hrs of artificial light at one time and 3 hours at another time. Which exposure time causes more blooms on average? Cars are filled with Fuel A at one time and Fuel B at another time. Which fuel gets better mpg on average?

11. Hypothesis Testing Using T Tests The t test allows a scientist to determine whether their research hypothesis is supported Null hypothesis H 0 : The IV does not influence the DV Any differences in average scores between the different conditions are probably just due to chance (measurement error, random sampling error)

12. Hypothesis Testing Using T Tests Research hypothesis H A : The IV does influence the DV The differences in average scores between the different conditions are probably not due to chance but show a real effect of the IV on the DV  Teaching tip: Very important for students to be able to understand and state the research question so that they see that statistics is a tool to answer that question

13. Hypothesis Testing Using T Tests Null hypothesis: Average pain relief is the same whether people have Drug A or a placebo Research hypothesis: Drug A brings better pain relief on average than the placebo (Upper tail test) Null hypothesis: Plants exposed to 3 hours of artificial light have the same number of blooms on average as plants not exposed to artificial light Research hypothesis: Plants exposed to 3 hours of artificial light have a different number of blooms on average than plants not exposed to artificial light (Two tailed test) Null hypothesis: Cars filled with Fuel A get the same mpg as Fuel B on average Research hypothesis: Cars filled with Fuel A get less mpg than Fuel B on average (Lower tail test)

14. Hypothesis Testing Using T Tests Do the data/evidence support the research hypothesis or not? Did the IV really influence the DV, or are the obtained differences in averages between conditions just due to chance? Teaching tip: To increase your students’ understanding, you should be explicit about what researchers mean by “just by chance”…

15. Hypothesis Testing Using T Tests p value = probability of results being due to chance When the p value is high (p >.05), the obtained difference is probably due to chance.99.75.55.25.15.10.07 When the p value is low (p <.05), the obtained difference is probably NOT due to chance and more likely reflects a real influence of the IV on DV.04.03.02.01.001

16. Hypothesis Testing Using T Tests p value = probability of results being due to chance [Probability of observing your data (or more severe) if H 0 were true] When the p value is high (p >.05), the obtained difference is probably due to chance [Data likely if H 0 were true].99.75.55.25.15.10.07 When the p value is low (p <.05), the obtained difference is probably NOT due to chance and more likely reflects a real influence of the IV on DV [Data unlikely if H 0 were true, so data support H A ].04.03.02.01.001

17. Hypothesis Testing Using T Tests In science, a p value of.05 is a conventionally accepted cutoff point for saying when a result is more likely due to chance or more likely due to a real effect Not significant = the obtained difference is probably due to chance; the IV does not appear to have a real influence on the DV; p >.05 Statistically significant = the obtained difference is probably NOT due to chance and is likely due to a real influence of the IV on DV; p <.05

18. Types of T Tests All 3 t tests in essence answer the same question: Is the research hypothesis supported? In other words, did the IV really influence the DV, or are the obtained differences between conditions just due to chance? One sample t test Independent samples t test Paired samples t test

19. The Essence of a T Test They answer this question by calculating a t score The t score basically examines how large the difference between the average score in each condition is, relative to how far spread out you would expect scores to be just based on chance (i.e., if there really was no effect of the IV on the DV) It is of course more sophisticated than that but students will have to wait for statistics courses to really get the low down on how it all works

20. The Essence of a T Test This can be taught without formulas but for the sake of thoroughness… All formulas take the basic format: Difference between means Standard Error (SE) Remember: When you sample from a population, you won’t get the same exact mean every time. SE = how far you would expect sample means from the same population to vary due to chance

21. Formulas for T Tests One-sample t test: difference between sample mean and population mean divided by estimated standard error of the mean Independent samples t test: difference between sample means divided by standard error of the difference between means Related samples t test: mean of the difference scores – expected mean difference divided by standard error of the mean difference scores

22. The Essence of a T Test Each t test gives you a t score, which can be positive or negative; It’s the absolute value that matters The bigger the |t| score, the less likely the difference between conditions is just due to chance The bigger the |t| score, the more likely the difference between conditions is due to a real effect of the IV on the DV So big values of |t| will be associated with small p values that indicate the differences are significant (p <.05) Little values of |t| (i.e., close to 0) will be associated with larger p values that indicate the differences are not significant (p >.05)

23. Each t test will also have a particular value for degrees of freedom (df), which is based on the sample size (N) Excel will calculate this for you, based on the sample size. The test statistic and df are both needed to calculate the p-value. Testdf formulaWhat it measures One sample t N – 1 total number of subjects minus 1 Independent samples t (Pooled) (N 1 – 1)+(N 2 – 1) (number of subjects in Cond. 1) minus 1 + (number of subjects in Cond. 2) minus 1 Paired samples t N – 1 total number of subjects minus 1 * Remember, they are the same subjects in both conditions so do not count them twice

24. What Software to Use to Analyze Your Data? If you have raw data, you can use Excel or SPSS If the data has already been summarized (e.g., you do not have the raw data but just have group means and SDs), you need to use Excel

25. Running the One-Sample T Test Using Excel One Sample t Test: Use when you have one sample and you want to compare it to a known population Need: Descriptive statistics: Sample M and SD Known population mean (to compare sample to) To run: Open Excel file “SFE Statistical Tests” and go to page called One-sample t test Enter population mean (mu or  ), sample size (N), sample mean (M), and standard deviation (SD) Output: Computer calculates t value, df, and p value

26. Running the Independent Samples T Test Using Excel Independent samples t Test: Use when you have a between- subjects design Need: Descriptive statistics: M, SD, and N for each condition To run: Open Excel file “SFE Statistical Tests” and go to page called Independent samples t test Pooled: If population SDs are equal Not Pooled: If population SDs are not equal Enter for each condition the sample size (N), Mean (M), and standard deviation (SD) Output: Computer calculates t value, df, and p value

27. Running the Paired Samples T Test Using Excel Paired samples t Test: Use when you have a within- subjects design Need: Descriptive statistics: Difference scores for the paired data (X-Y); M and SD for the difference scores To run: Open Excel file “SFE Statistical Tests” and go to page called Paired samples t test Enter the sample size (N), Mean (M) and standard deviation (SD) of the difference scores Output: Computer calculates t value, df, and p value

28. Running the One-Sample T Test Using SPSS One Sample t Test: Use when you have one sample and you want to compare it to a known population Need: Raw numerical data in a column Known population mean (to compare sample to) To run: Open SPSS; Analyze  Compare means  One sample t test; send the column with your raw data into the “Test variable” box; Under “Test value” enter the known population mean, hit Ok Output: Computer calculates t value, df, and p value

29. Running the Independent Samples T Test Using SPSS Independent samples t Test: Use when you have a between- subjects design Need: Raw numerical data set up in two columns (IV; DV); make sure you use value labels under IV to indicate what the #s stand for (e.g., 1=male, 2=female) To run: Analyze  compare means  independent samples t test Test var: DVGrouping var: IV (condition) Define groupsGroup 1: 1 Group 2: 2, ok Output: Computer calculates t value, df, and p value

30. Running the Paired Samples T Test Using SPSS Paired samples t Test: Use when you have a within- subjects design Need: Set up data file as two columns listing raw numerical data: Cond 1, Cond 2 [but give meaningful names to] To run: Analyze  compare means  paired samples t test; send pair over Output: Computer calculates t value, df, and p value

31. Interpreting T Tests Cardinal rule: Scientists do not say “prove”! Conclusions are based on probability (likely due to chance, likely a real effect…). Be explicit about this to your students. Based on p value, determine whether you have evidence to conclude the difference was probably real or was probably due to chance: Is the research hypothesis supported? p <.05: Significant Reject null hypothesis and support research hypothesis (the difference was probably real; the IV likely influences the DV) p >.05: Not significant Retain null hypothesis and reject research hypothesis (any difference was probably due to chance; the IV did not influence the DV)

32. Teaching Tips Students have trouble understanding what is less than.05 and what is greater, so a little redundancy will go a long way! Whenever you say “p is less than point oh-five” also say, “so the probability that this is due to chance is less than 5%, so it’s probably a real effect.” Whenever you say “p is greater than point oh-five” also say, “so the probability that this is due to chance is greater than 5%, so there’s just not enough evidence to conclude that it’s a real effect – these 2 conditions are not really different” In other words, read the p value as a percentage, as odds, “the odds that this difference is due to chance are 1%, so it’s probably not chance…” Relate your phrasing back to the IV and DV: “So the IV likely caused changes in the DV…”; “The IV worked – these 2 conditions are different”

33. Deciding to Use One-tailed or Two-tailed T Tests Different disciplines have different conventions The conservative “rule” in some disciplines is to use a two-tailed test, regardless of whether you had a directional or nondirectional hypothesis Directional: You expect scores in one condition to be higher than the other Nondirectional: You expect a difference between conditions but do not specify ahead of time whether A should be greater than B or less than B If using two-tailed, instruct your students to look for the p value on the line that says “two tailed” in the Excel worksheet

34. Deciding to Use One-tailed or Two-tailed T Tests Example: For the two sample tests (both paired and independent tests) Null Hypothesis:  1 =  2 Research Hypothesis:  1   2 (Two tail test) or  1 >  2 (Upper tail test) or  1 <  2 (Lower tail test) Where  1 is the mean of the first population and  2 is the mean of the second population

35. Reporting T Tests Results State key findings in understandable sentences Use descriptive and inferential statistics to supplement verbal description by putting them in parentheses and at the end of the sentence Use a table and/or figure to illustrate findings

36. Reporting T Tests Results Step 1: Write a sentence that clearly indicates what statistical analysis you used A [type of t test] t test was conducted to determine whether [name of DV] varied as a function of [name of IV or name of conditions] A one-sample t test was conducted to determine whether the pollen count varied as a function of location (on campus or in the state of CT in general) An independent samples t test was conducted to determine whether people’s pulse rates varied as a function of their weight (obese people or underweight). A paired samples t test was conducted to determine whether calories consumed by rats varied as a function of group size (rats tested alone, tested in groups).

37. Reporting T Tests Results Step 2: Write a sentence that clearly indicates what pattern you saw in your data analysis – Did the conditions differ, and if so, how (i.e., which condition scored higher or lower?) Average [Name of DV] was significantly higher/lower for [name of Condition 1] than the average for [name of Condition 2] [Name of DV] was significantly higher/lower on average for [name of Condition 1] than for [name of Condition 2] [Name of DV] did not significantly differ between [name of Condition 1] and [name of Condition 2] on average The pollen count was significantly lower on average on college campuses than in the state of CT in general. (Lower tailed test) Pulse rates did not significantly differ on average whether people were obese or underweight. (Two tailed test) The number of calories on average consumed by rats was significantly higher when the rats ate alone than when they ate in groups. (Upper tailed test)

38. Teaching Tip The key is to emphasize that these should be nice, easy-to-understand grammatical sentences that do not sound like “Me Tarzan, you Jane!” You may want your students to explicitly note whether the research hypothesis was supported or not. “Results supported the hypothesis that increased dosages of the drug would reduce the average number of white blood cells…”

39. Reporting T Tests Results Step 3: Tack the descriptive and inferential statistics onto the sentence Put Ms and SDs in parentheses after the name of the condition. (Possibly include Ns for two sample tests.) Put the t test results at the end of the sentence using this format: t(df) = x.xx, p =.xx The pollen count was significantly lower on average on college campuses (M=8.35, SD=2.25) than in the state of CT in general (  =9.85), t(12)= -2.40, p =.02. Pulse rates did not significantly differ on average whether people were obese (M=95 beats per minute, SD=21, N = 24) or underweight (M=91 beats per minute, SD=18, N = 23), t(45)= 0.70, p =.49. The number of calories on average consumed by rats was significantly higher when the rats ate alone (M=100.34, SD = 12.64) than when they ate in groups (M=87.65, SD = 13.43), t(22) = 2.38, p =.01.

40. Reporting T Tests Results for Nonsignificant Difference Teaching tip: Make sure your students understand that when the difference is not significant, they should NOT word their sentences to imply that there is a difference Not significant = no difference One mean will no doubt be higher than the other, but if it’s not a significant difference, then the difference is probably not real, so do not interpret a direction (Saying “this is higher but no it’s really not” is silly) Option 1: Simply say there was no significant difference Pulse rates did not significantly differ on average whether people were obese (M=95 beats per minute, SD=21, N = 24) or underweight (M=91 beats per minute, SD=18, N = 23), t(45)= 0.70, p =.49. Option 2: Word the sentence so that it is clear that the hypothesized difference was not obtained Counter to hypotheses, pulse rates were not significantly higher on average for people who were obese (M=95 beats per minute, SD=21, N = 24) than for people who were underweight (M=91 beats per minute, SD=18, N = 23), t(45)= 0.70, p =.49.

41. Reporting T Tests Results Be sure that you note the unit of measure for the DV (miles per gallon, volts, seconds, #, %). Be very specific If using a Table or Figure showing M & SDs or SEs, you do not necessarily have to include those descriptive statistics in your sentences If you have more than 2 conditions, run t tests for each pair of conditions you want to compare (e.g., Cond. 1 vs. 2, Cond 1 vs. 3, Cond. 2 vs. 3)

42. More Teaching Tips You can ask your students to report either: the exact p value (p =.03, p =.45) the cutoff: say either p.05 (not significant) You should specify which style you expect. Ambiguity confuses them! Tell students they can only use the word “significant” only when they mean it (i.e., the probability the results are due to chance is less than 5%) and to not use it with adjectives (i.e., they often mistakenly think one test can be “more significant” or “less significant” than another). Emphasize that “significant” is a cutoff that is either met or not met -- Just like you are either found guilty or not guilty, pregnant or not pregnant. There are no gradients. Lower p values = less likelihood result is due to chance, not “more significant”

43. Effect Size for T Test When a result is found to be significant (p <.05), many researchers report the effect size as well Effect size = How large the difference in scores was Significant = Was there a real difference or not?

44. Effect Size for T Test Effect size: How much did the IV influence the DV? Cohen’s d = |mean difference| SD Small0 -.20 Medium.21 -.80 Large >.80

45. Reporting T Tests Results ****THIS STEP IS OPTIONAL*** Step 4: Report the effect size if the t test was significant After the t test results are reported, say whether it was a small, medium, or large effect size, and report Cohen’s d The pollen count was significantly lower on average on college campuses (M=8.35, SD=2.25) than in the state of CT in general (  =9.85), t(12)= -2.40, p =.02, and the effect size was medium, Cohen’s d =.67 Pulse rates did not significantly differ on average whether people were obese (M=95 beats per minute, SD=21, N = 24) or underweight (M=91 beats per minute, SD=18, N = 23), t(45)=.70, p =.49. No reason to report effect size because t test was not significant. The number of calories on average consumed by rats was significantly higher when the rats ate alone (M=100.34, SD = 12.64) than when they ate in groups (M=87.65, SD = 13.43), t(22) = 2.38, p =.01, and the effect size was large, Cohen’s d =.97.

46. Check Assumptions for T Test Numerical scale (interval or ratio) for DV The distribution of scores in each sample is approximately symmetrical (normal) thus the mean is an appropriate measure of central tendency If a distribution is somewhat skewed (not symmetric) it is still acceptable to run the the test as long as the sample size is not too small (say, N > 30)

47. Time to Practice Running and reporting t tests