# Can I do Paired T-test for averaged data

I will make my question more obvious here. My data consists of each students making 5 attempts to say, throw a ball in a hole back to back. My interest is to see, how many times they succeed, whether the average speed of throwing differs between passed and failed attempts. So I will also have the measurement on the speed of throwing. I know I can do a repeated measure ANOVA to test the mean difference of speed between throws, but that's not my intention. I was wondering if I could do a pair T-test to see the mean difference between successful and unsuccessful attempts. Can I take the average of the successful and unsuccessful attempts for each student and run a paired T-test (if normality is satisfied) on these data? Any suggestion is appreciated! Let me know as well if you have any questions. Thanks!

• What would make an appropriate pairing? – Michael R. Chernick Jan 23 '17 at 22:16
• @MichaelChernick Assume me as a student made 5 attempts. I failed 3 times, passed 2 times. I can calculate the average speed of throwing for my successful attempts, as well as for my failed attempts. These 2 averages will constitute my pair. Does it make sense? – curiousmind Jan 23 '17 at 22:28
• Why isn't it your "intention" to do a repeated measure ANOVA? – StatsStudent Jan 23 '17 at 22:31
• How would you pair if all attempts where passed or all failed? – StatsStudent Jan 23 '17 at 22:33
• A paired t test involves a set of paired correlated observations. What is your objective? – Michael R. Chernick Jan 23 '17 at 22:57

The paired t-test should be used where a single condition changes within a setting that otherwise remains the same. For example, blood pressure measurements within the same individual before and after starting a medicine.

Here you seem to be interested in the number of successful attempts at throwing a ball into a hole, and whether the speed of the throw affects the probability of success. The measurements are not independent because each individual throws five times (and I am ignoring any practice effect where the accuracy would improve from the first to the fifth throw).

A simple representation would be a two level model with a logistic link so the outcome is 1 for success and 0 for failure, and the explanatory variables include the measure of speed, and a dummy variable for the individual (to handle the within individual correlation).

• Thanks! The reason I picked paired T-test (without realizing the challenges it poses) was this is the simplest way to test the difference. But the regression model you proposed sounds good. So my regression model would be: y = \beta_{1}. x1 + \beta_{2}*x2 + e; where y = success or failure, x1= individuals (ie. 1,1,1,1,1 for the 1st student), x2= measure of speed. So the data look like a long data. Am I getting it right? Thanks. – curiousmind Jan 24 '17 at 16:12
• One thought just crossed my mind. Whether it would be okay to include a dummy variable for the individuals as a regressor in the model, or defining this variable as a clustering variable looks more convincing? – curiousmind Jan 24 '17 at 21:34
• It depends! If you're interested in the specific individuals then a dummy variable would give you insight, but if you have lots of individuals that are just providing more data to understand the effect of speed better then include the individuals as a random effect. – drstevok Jan 24 '17 at 22:01
• And your model looks OK assuming you are including the link function – drstevok Jan 24 '17 at 22:02

You can compare the speeds of the successful and unsuccessful throws and do so with a design that recognizes pairing, but there are challenges you need to address. First, you may have missing data, as when one person misses all the throws. Second, the averages will have a smaller variance than individual throws, and the variance will be smaller when you average more throws. Different variances for different observations violates one of the assumptions for a t-test (paired or not). So, you might consider a non-parametric statistical test.

• Thanks! I didn't realize doing so will violate a basic assumption. Thank you for bringing it up. – curiousmind Jan 24 '17 at 16:00
• I'll just toss in this great article that talks about the dangers of using ANOVA with this kind of data. It also talks about using mixed effect logistic regression, which might be a way to go! ncbi.nlm.nih.gov/pmc/articles/PMC2613284 – Dave Aug 9 '17 at 22:04