Please enable JavaScript.
Coggle requires JavaScript to display documents.
The t Test for Two Related Samples - Coggle Diagram
The t Test for Two Related Samples
repeated-measures design
: one group of participants is measured in two different treatment conditions - there are two scores for each individual in the sample
the repeated- measures t is based on difference scores rather than raw scores
difference scores:
obtained by subtracting the first score (before treatment) from the second score (after treatment) for each individual
D = X2-X1
D tells us the direction of change
the D scores serve as the sample data for the hypothesis test
to calculate the t statistic - use the number of D scores (n) as well as the mean for the sample of D scores (MD) and the value of SS for the sameple of D scores
hypotheses for a repeated-measures t test
we are interested in a population of difference scores
more specifically, the mean for the population difference scores
the null hypothesis states that the mean difference for the general population is zero
although population mean might be zero, the individual scores are not all equal to zero - some individuals will have positive D scores and some will have negative D scores
repeated-measures t statistic
: t = (MD - µD)/ estimated standard error of D
variance:
s² = SS/(n-1) or s = √SS/df
estimated standard error:
estimated standard error = √s²/n or s/√n
hypothesis test for the repeated-measures design
: we are interested in whether or not there is a systematic difference between the scores in the first treatment and the scores in the second treatment
step 1: state the hypothesis and select an alpha level (.05)
step 2: locate the critical region
step 3: calculate the t statistic
step 4: make a decision
assumptions of the related samples t test:
1) the observations
within
each treatment condition must be independent
2) the population distribution of difference scores must be normal
matched-subjects design
: involves two separate samples, but each individual in one sample is matched one-to-one with any individual in the other sample
matched with respect to a specific variable
has the advantages of both independent-measures and repeated-measures design without the disadvantages of either one
effect size
most commonly used methods are Cohen's d and r²
estimated d = sample mean difference/sample standard deviation
r² = the same as previous formulas
r² = t²/(t²+df)
impacted by variance, but only a very small amount by sample size
impacted by variance, but not by sample size
confidence intervals for estimating population mean difference
µD = MD +/- t(estimated standard error)
influenced by:
1) size of treatment effect
2) width of interval depends on percentage of confidence used (larger confidence = wider interval)
2) larger sample size = more narrow interval
because interval width is related to sample size, it is not a pure measure of effect size like Cohen's d or r²
sample variance and sample size
standard error is inversely related to sample size and directly related to variance
a bigger sample produces a larger t statistic and increases the likelihood of rejecting H0
larger variance = smaller value for the t statistic and reduces the likelihood of finding a significant result
variability describes the consistency of the treatment effect
repeated - and independent-measures designs
repeated-measures typically requires fewer subjects
repeated-measures are great for looking at changes over time
repeated-measures eliminates problems caused by individual differences
repeated-measures tends to have more power than an independent-measures
lower values for standard error result in greater statistical power
a disadvantage of the repeated-measures design is that the structure allows for external factors outside of the treatment effect to cause a participant's score to change from one treatment to the next
if there is a strong chance of time effecting the effects, it's best to just do do an independent measures design
another possible disadvantage of the repeated-measures design is that it is also possible that participation in the first treatment influences the individual's score in the second treatment which is called
order effect
one way to address this issue is to counterbalance the order of presentation of treatments