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t-Test. t-Test. Two groups of random samples from sample population prior to experiment: group 1 = group 2 any differences can be attributed to: sampling error chance. t-Test. Following the experiment, difference can be attributed to: treatment effect - (explained variance)
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t-Test • Two groups of random samples from sample population • prior to experiment: group 1 = group 2 • any differences can be attributed to: sampling error chance
t-Test Following the experiment, difference can be attributed to: • treatment effect - (explained variance) • error variance - (unexplained variance)
Error Variance • measurement error • environmental factors “People are different” • personal characteristics • behavioral factors
Statistical Ratio t ratio = difference between group means variability within groups = rx effect + error error
Statistical Ratio if there is a rx effect: = rx effect + error error t-ratio is relatively large
Statistical Ratio • if there is NO rx effect or the effect is small then… rx effect + error error approaches error/error
Statistical Ratio • t-ratio is relatively small (--> 1) • and H0 is true
Between/Within Variance “variance between” • variance between the groups • difference between means “variance within” • variance within groups
Between/Within Variance To show difference (statistical significance) you want: • variance between to be relatively large • variance within to be relatively small
Assumptions • randomly sampled groups • normally distributed groups • homogeneity of variance/homoscedasticity
Violations of Assumptions randomly sampled groups • can’t normality • research has demonstrated almost no practical consequences w/ t-test
Violations of Assumptions Homogeneity of variance • if n1 = n2 --> not a problem • if n1 n2 --> can be a problem test homogeneity of variance w/: Levene’s test Bartlett’s test
Violations of Assumptions Homogeneity of variance • only an issue w/ independent t-tests • wait a few minutes
t-Test Basics Comparison two groups • independent • dependent
Independent t-Test • Unpaired • groups are independent - no relationship
Independent t-Test Effect of one week of hand splinting on pinch strength in subjects w/ RA: • treatment group • control group • pre-post design • difference scores
Independent t-Test Check assumptions: • random samples • normality • homogeneity of variance?
Independent t-Test Homogeneity of variance: • If (n1 = n2) --> not a concern • However, can check w/: Levene or Bartlett test • If variances significantly different (p > .05) --> t-test for equal variances • else t-test for unequal variances
Independent t-Test Why do all this? • Determines what t-statistic formula to use and degrees of freedom (DOF) • t-test for equal variances is more powerful
Independent t-Test • Look up table • for t-ratio to be sign. the t-ratio critical value • ( = .05)t(18) = 1.734 (one-tailed) • which way?
Independent t-Test • ( = .05)t(18) = 2.101 (two-tailed) • less powerful (2.101 > 1.734)
Independent t-Test Computer programs : • report t-statistic • exact p values (p < 0.05)
Dependent t-Test Repeated measure (pre-post of same group) matched design • twins • matched for age, gender, etc.
Dependent t-Test Effect of aquatic therapy on balance Assumptions: • random sample • normality • homogeneity of variance
Multiple t-Tests • BAD!!!!!!!! • Multiple comparisons on the same sample • More likely to make a type I error rx effect + error error
Multiple t-Tests • Run a different analysis (ANOVA) • adjust = by dividing by # of tests
Analysis of Variance (ANOVA) Similar to t-test but: • compares 3 or more conditions or groups F-statisitic = between-group differences within-group variability
ANOVA Can analyze: • independent samples (unpaired t) • dependent samples (paired t) • combination
One-way ANOVA • Single factor or independent variable w/ multiple levels • 3 or more independent groups are compared
One-way ANOVA Effect of assistive devices on step length: • cane • crutches • control Cane Crutches Control
Two-way ANOVA • Two independent variables or factors • Each IV has multiple levels • Effect of type of stretch and knee position on increasing ankle ROM
Two-way ANOVA Two independent variables (factors): • Stretch type (Factor A) • Knee position (Factor B)
Two-way ANOVA Stretch type (Factor A) has 3 levels: • prolonged • quick • none (control) Knee position (Factor B) has 2 levels: • Flexed • Extended
Two-way ANOVA Stretch type (Factor A) has 3 levels Knee position (Factor B) has 2 levels 3 X 2 Factorial Design
Two-way ANOVA Stretch (Factor A) Knee Position (Factor B)
Two-way ANOVA - Main Effects Stretch (Factor A) Knee Position (Factor B)
Two-way ANOVA - Main Effects Stretch (Factor A) Knee Position (Factor B)
Interactions • Effects of one variable are not constant (different) across different levels of a 2nd variable • The effect of stretching on ankle ROM works differently depending on the position of the knee.
