t-statistics

Prob. is the two-sided probability of the absolute value of the reported t with the reported d.f., so that a statistically significant finding for a single two-sided t-test is Prob.<.05, and a highly significant finding is Prob.<.01. Please interpret this conservatively, i.e., "barely significant is probably not significant". If you wish to make a Bonferroni multiple-comparison correction, compare this Prob. with your chosen significance level, e.g., p<.05, divided by the number of DIF tests in this Table. This is approximate because of dependencies between the statistics underlying the computation.

 

Many statistical tests are reported as Student's t statistics. This table shows the significance-level values for different degrees of freedom (d.f.). Often the reported t-statistics have effectively infinite degrees of freedom and so approximate a unit normal distribution. t-statistics with infinite degrees of freedom are also called z-statistics, paralleling the use of "z" in z-scores.

 

 

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Quick approximate degrees of freedom for 2-sample t-test: d.f. = size of sample 1 + size of sample 2 - 2

 

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Welch's refinement of Student's t-test for possibly unequal variances of two samples:

 

For sample 1,

M1 = mean of the sample

SS1 = sum of squares of observations from the individual sample means

N1 = sample size (or number of observations)

SS1 / (N1 - 1) = sample variance around the mean (or the measure)

SS1 / ((N1 - 1)(N1)) = standard error variance = EV1 = SE1²

SE1 = Sqrt(EV1) = standard error of the mean (or the measure)

 

Similarly for sample 2, then

t = (M1 - M2) / sqrt (EV1 + EV2) = (M1 - M2) / sqrt (SE1² + SE2²)

 

with Welch-Satterthwaite d.f. = (EV1 + EV2)² / (EV1²/ (N1-1) + EV2² /(N2-1))

which is the same as d.f = (SE1² + SE2²)² / ( (SE14 / (N1-1)) + (SE24 / (N2-1)))

 

A calculator for this is at https://www.graphpad.com/quickcalcs/ttest1.cfm?Format=SEM

 

Satterthwaite, F. E. (1946), "An Approximate Distribution of Estimates of Variance Components.", Biometrics Bulletin 2: 110-114  

Welch, B. L. (1947), "The generalization of "Student's" problem when several different population variances are involved.", Biometrika 34: 28-35

 

Example: Gender subtotals for Example0.txt Table 28:

M1 = 1.62, M2 = .76, SE1 = .38, SE2 = .16, N1 = 18, N2 = 57

Welch: t = 2.08, d.f. = 23, p = .049