text: ﻿ t-statistics
  
  # 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.
  
  Table of the two-sided t distribution
  
   d.f. p=.05 p=.01 1 12.71 63.66 2 4.30 9.93 3 3.18 5.84 4 2.78 4.60 5 2.57 4.03 6 2.45 3.71 7 2.37 3.50 8 2.31 3.36 9 2.26 3.25 10 2.23 3.17
   d.f. p=.05 p=.01 11 2.20 3.11 12 2.18 3.06 13 2.16 3.01 14 2.15 2.98 15 2.13 2.95 16 2.12 2.92 17 2.11 2.90 18 2.10 2.88 19 2.09 2.86 20 2.09 2.85
   d.f. p=.05 p=.01 21 2.08 2.83 22 2.07 2.82 23 2.07 2.81 24 2.06 2.80 25 2.06 2.79 30 2.04 2.75 100 1.98 2.63 1000 1.96 2.58 Infinite 1.96 2.58 (z-statistic)
  
  A calculator for the probability of any t value and d.f. is at http://www.danielsoper.com/statcalc3/calc.aspx?id=8
  
  Welch's refinement of Student's t-test for possibly unequal variances:
  
  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 http://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
  
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