dependent t-tests

Factors affecting statistical power in the t-test Statistical power ability to identify a statistically significant difference when a difference between means actually exists

Decision Table: Correct DECISION REALITYREALITY Truth is everlasting, but our ideas about truth are interchangeable

Factors affecting statistical power in the t-test  level how much risk are YOU willing to take in making a Type I error Frank & Huck (1986, RQES): Why does everyone use the 0.05 level of significance? 0.01 conservative 0.10 liberal Power

Factors affecting statistical power in the t-test  level df (number of subjects) affects variability associated with the sample mean & variability within the sample limited by time & money GREATER n = GREATER POWER (point of diminishing return)

Statistics Humour One day there was a fire in a wastebasket in the Dean's office and in rushed a physicist, a chemist, and a statistician. The physicist immediately starts to work on how much energy would have to be removed from the fire to stop the combustion. The chemist works on which reagent would have to be added to the fire to prevent oxidation. While they are doing this, the statistician is setting fires to all the other wastebaskets in the office. "What are you doing?" they demanded. "Well to solve the problem, obviously you need a large sample size" the statistician replies.

Factors affecting statistical power in the t-test  level df (number of subjects) magnitude of the mean difference how different are the treatments imposed measurement errors sampling errors SIZE OF THE TREATMENT EFFECT

Factors affecting statistical power in the t-test  level df (number of subjects) magnitude of the mean difference variability how specified is your population control of extraneous variables

Estimated Standard Error of the Difference between 2 independent means

t-test for independent samples Smaller is better

Comparing paired (correlated) measures instead of group (uncorrelated) measures Match subjects what factors (variables) might affect time to exhaustion on the exercise bike daily diet? Fitness level? Genetics? Height? Weight? Age? Regular training program?

Comparing paired (correlated) measures instead of group (uncorrelated) measures Match subjects Repeated measures measure the SAME subject under both protocols test & retest pre & posttest condition 1 & condition 2

Comparing paired (correlated) measures instead of group (uncorrelated) measures Match subjects Repeated measures Subject serves as own Control

Comparing paired (correlated) measures instead of group (uncorrelated) measures Match subjects Repeated measures Subject serves as own Control Intra-subject variability should be LESS than Inter-subject variability

Dependent t-test (paired or correlated t-test) Pairs of scores are matched same subject in 2 conditions or matched subjects Question: Does ankle bracing affect load during landing? IV: brace condition DV: Vertical GRF

Steps to dependent t-test Set  (0.05) Set sample size One randomly selected group n = 7 condition 1: Brace condition 2: No brace Set H o (null hypothesis)

Set statistical hypotheses H o Null hypothesis Any observed difference between the two conditions will be attributable to random sampling error. H A Alternative hypothesis If H o is rejected, the difference is not attributable to random sampling error perhaps brace???

Steps to independent t-test Set  (0.05) Set sample size (n = 7) Set H o Test each subject in both conditions with a standardized protocol (drop landings) Note: condition performance order is randomized across subjects

GRF data

Steps to dependent t-test Set  (0.05) Set sample size (n = 7) Set H o Test each subject in both conditions Calculate descriptive statistics of each condition scattergram mean, SD, n

Figure 1. Scattergram of vertical GRF during landing in different brace conditions (N/kg)

Descriptive statistics for atble401.sav data

Steps to dependent t-test Set  (0.05) Set sample size (n = 7) Set H o Test each subject in both conditions Calculate descriptive statistics of each condition compare the condition means

How to compare the condition means Even if the two conditions were the same (samples drawn from the same population), would not expect the statistics to be the same Need a measure of expected variability against which the mean of the difference between paired scores (X i - Y i ) could be compared

Paired scores, so the data are somewhat correlated Calculate the difference between the two conditions for each case (X i - Y i ) Calculate the Mean Difference Use the correlation among the pairs of scores to reduce the error term (denominator) used to evaluate the difference between the means

t-test for dependent (paired) samples t = M diff SE MD

GRF data  = -20 Mean Diff = -2.9

t-test for dependent (paired) samples t = M diff SE MD Standard error of the Mean difference for Paired Scores

Estimated Standard Error of the Difference between 2 dependent means ?

If r = 0, this term reduces to the same equation as for independent groups

t-test for dependent (paired) samples t = M diff SE MD df = ??

t-test for dependent (paired) samples t = M diff SE MD df = n pairs - 1

Running the dependent t-test with SPSS Enter the data as pairs atble401.sav

Reporting paired t-test outcome Table 1. Descriptive statistics of vertical ground reaction force (in N/kg) for the two conditions (n = 7)

Reporting t-test outcome * Figure 1. Mean vertical GRF in the two conditions (* p  0.05)

Reporting t-test in text Descriptive statistics of the vertical ground reaction force (VGRF) data during landing in the two braced conditions are presented in Table 1 and graphically in Figure 1. A paired t-test indicated that the mean VGRF of 10.9 (SD = 3.5) N/kg in the braced condition was significantly higher (  = 0.05) than the mean VGRF of 8.0 (4.3) N/kg in the unbraced condition (t 6 = 3.57, p = 0.012). The mean difference of 2.9 N/kg represents a 36% higher VGRF during the landings with a brace compared to without a brace.

What if you set  = 0.01? Descriptive statistics of the vertical ground reaction force (VGRF) data during landing in the two braced conditions are presented in Table 1 and graphically in Figure 1. A paired t-test indicated that the mean VGRF of 10.9 (SD = 3.5) N/kg in the braced condition was...

What if you set  = 0.01? Descriptive statistics of the vertical ground reaction force (VGRF) data during landing in the two braced conditions are presented in Table 1 and graphically in Figure 1. A paired t-test indicated that the mean VGRF of 10.9 (SD = 3.5) N/kg in the braced condition was significantly higher (  = 0.01) than the mean VGRF of 8.0 (4.3) N/kg in the unbraced condition (t 6 = 3.57, p = 0.012). The mean difference of 2.9 N/kg represents a 36% higher VGRF during the landings with a brace compared to without a brace. not

Statistics Humour A student set forth on a quest To learn which of the world’s beers was best But his wallet was dried out At the first pub he tried out With two samples he flunked the means test Gehlbach, SH (2002) Interpreting the medical literature

Summary: both t-tests are of the form: t = Standard Error Mean Difference

To increase statistical power t = Standard Error Mean Difference Maximize Minimize

Choosing which t-test to use Independent no correlation between the two groups Dependent two sets of data (pair of scores) from matched subjects or from the same subject (repeated measures) data are correlated

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