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1 INSTRUCTIONS CLICK LEFT SIDE OF MOUSE TO GO FORWARD (or PageDown key) CLICK RIGHT SIDE OF MOUSE TO GO BACK (or PageUp key) Press ESC key (top left of keyboard) to QUIT at any time A CAL COMPANION TO EXCEL PRINT-OUTS MP110 SKILLS FOR LIFE SCIENCES Lecture 2 : t Tests These notes are the intellectual property of Dr M E Jakobson in support of his lectures and are solely for bona fide use for study by students registered on courses at UeL. NO other use is permitted without prior permission.

2 MP110 TWO REVISION LECTURES ON STATISTICS Dr Mike Jakobson Lecture 1 : INITIAL EXCEL DATA DESCRIPTION before COMPARING SAMPLES Lecture 2 : EXCEL t TESTS INTERPRETATION when COMPARING SAMPLES

3 Look back at statistics covered so far on MP110 Look forward by using examples of data analysis done by Level 2 students in practicals in PP249 Physiological Function & Dysfunction PP250 Physiological Regulation

4 OUTCOMES : you should be better able to distinguish between correct and incorrect statements about EXCEL t TESTS Lecture 2 : EXCEL t TESTS INTERPRETATION when COMPARING SAMPLES

5 HANDOUT in YOUR MP110 WORKBOOK Also files on INTRANET DEPARTMENT MENU Directory : Skills for Life Sciences MPA100 Semester B data handling MPA110_MJSTATS FILE ONE_QUESTIONS.doc MPA110_MJSTATS FILE TWO_ANSWERS.doc MP110_MJSTATS LECTURE 1.pps MP110_MJSTATS LECTURE 2.pps

6 also see.. MP110 WebCT SELF-TEST MCQ (Q11-Q22) on your EXCEL print-outs

7 EXCEL t TESTS t Stat P value Hypothesis= NULL HYPOTHESIS (N.H.) Hypothesized Mean Difference = 0 The bigger the t value, the greater the mismatch with the N.H. The smaller the P value, the more significant the difference from N.H.

8 EXCEL t TESTS What type of t test ? Paired sample or unpaired 2-sample ? One-tailed or two-tailed ? Assuming Equal or Unequal Variances ?

9 Q20In a study on B.P. on 8 males, systolic B.P. when standing (mean 124 mm Hg +/- S.D. 5) was compared in all 8 with their values when supine (mean 118 mm Hg +/- S.D. 4). The most appropriate t test to apply is: A a 2-tailed two-sample t test assuming equal variances B a 2-tailed paired sample t test C a 2-tailed two-sample t test assuming unequal variances D a 1-tailed paired sample t test E a 1-tailed two-sample test assuming equal variances alteration

10 Q20In a study on B.P. on 8 males, systolic B.P. when standing (mean 124 mm Hg +/- S.D. 5) was compared in all 8 with their values when supine (mean 118 mm Hg +/- S.D. 4). Two measurements taken on EACH male, use a PAIRED sample t test for MEAN CHANGE IN INDIVIDUAL B.P. Interested in change in EITHER direction (up or down) so use a TWO-TAILED t test

11 StandingSupineChange MEAN S.D. The paired t test looks at the 8 values for CHANGE N.H. could the MEAN be estimating a zero change?

12 Table 3 df NOT Df t not T lower case capital For Q13, Q14, Q15

13 Q13In a paired sample t test (e.g. Table 3) A the 18 subjects are assumed to be a random sample of the population B the mean value of the NORM sample is compared to the mean value of the DB:AIR sample C only 17 pairs of values are used as the degrees of freedom are 17 D the hypothesis assumes the variances of the two sets of data are unequal E the 18 values for NORM must be randomly paired with the 18 values for DB:AIR

14 Q13In a paired sample t test (e.g. Table 3) A the 18 subjects are assumed to be a random sample of the population IMPORTANT ASSUMPTION Each individual in the population equally likely to be sampled …...chance of being sampled not influenced by those already sampled

15 POPULATION being sampled Suppose a high value is randomly selected Then, the next value sampled is NOT affected in any way i.e. independent random sample next could be ANY of the other possibilities Random does NOT mean representative

16 t-Test: Paired Two Sample for Means NORM DB:AIR Mean Q13In a paired sample t test (e.g. Table 3) B the mean value of the NORM sample is compared to the mean value of the DB:AIR sample EXCEL TITLE MISGUIDED not PAIRED SAMPLE FOR MEANS, DATA VALUES are paired off not the MEANS. MEAN INDIVIDUAL CHANGE

17 t-Test: Paired Two Sample for Means NORM DB:AIR Mean Just one MEAN is being tested, namely the MEAN INDIVIDUAL CHANGE NORM - DB:AIR = MEAN INDIVIDUAL CHANGE

18 Q13In a paired sample t test (e.g. Table 3) C only 17 pairs of values are used as the degrees of freedom are 17 Degrees of freedom = N-1 18 individuals were in the sample, so there were 18 pairs of data

19 Table 3 df = degrees of freedom = (N - 1) N = number of values used in t test PAIRED t test USES NUMBER OF PAIRS so N=18 (not 36), and df = 18-1 = 17 the value used is the difference between each pair of observations

20 Q13In a paired sample t test (e.g. Table 3) D the hypothesis assumes the variances of the two sets of data are unequal Revision : calculate sample VARIANCE from S.D. ? from S.E.M. ? Variance = S 2 = S.D. N S.E.M. = S.D. N S.E.M. x (S.E.M.) 2 x N = Variance

21 Table 3 Variances are given in the print-out BUT in a PAIRED t TEST it does not alter validity of t value if they are different though it is still physiologically relevant

22 Q13In a paired sample t test (e.g. Table 3) E the 18 values for NORM must be randomly paired with the 18 values for DB:AIR But it is true that the INDIVIDUALS should be RANDOMLY CHOSEN from the population being studied NO! The whole idea is to look at change in performance WITHIN an individual

23 Q14 In Table 3, the t value (correct to 3 decimal places) calculated by the t test from the breath-holding data is : A B2.736 C1.740 D2.110 E Correctly rounded and without confusing extra.

24 Q15Assuming a two-tailed test and a 5% critical value, which of the following statements communicates the results of the paired t test in Table 3 accurately ? A There was no significant change in breath-holding (P<0.05) B There was a significant change in breath-holding (P=2.109) C There was a significant change in breath-holding (P<0.01) D There was no significant change in breath-holding (P=0.014) E There was a significant change in breath-holding (P<0.05) two-tailed test a difference in EITHER direction higher or lower can be regarded as significant So long as it is UNLIKELY to have occurred by chance i.e. P less than (<)5% (P<0.05)

25 Q15Assuming a two-tailed test and a 5% critical value, which of the following statements communicates the results of the paired t test in Table 3 accurately ? A There was no significant change in breath-holding (P<0.05) B There was a significant change in breath-holding (P=2.109) C There was a significant change in breath-holding (P<0.01) D There was no significant change in breath-holding (P=0.014) E There was a significant change in breath-holding (P<0.05) t Stat P(T<=t) two-tail t Critical two-tail P = probability that Null Hypothesis (of NO change) is TRUE t Stat = t value calculated from the OBSERVED data t Critical = largest t value (+ or -) EXPECTED from Null Hypothesis at = 0.05 (5% probability)

26 Q15Assuming a two-tailed test and a 5% critical value, which of the following statements communicates the results of the paired t test in Table 3 accurately ? A There was no significant change in breath-holding (P<0.05) B There was a significant change in breath-holding (P=2.109) C There was a significant change in breath-holding (P<0.01) D There was no significant change in breath-holding (P=0.014) E There was a significant change in breath-holding (P<0.05) t Stat P(T<=t) two-tail t Critical two-tail % students got this right Not a P value not <0.01, only <0.05 exact P values OK! wrong way round! is P OK?

27 Table 4 Q16 and Q18

28 Q16The t-Test Two-Sample assuming Equal Variances test is used here to test whether: Aone-tail tests or two-tail tests are better to use in your hypothesis Bthe two samples have significantly different sample sizes Cbreath-holding can be held for significantly longer than zero seconds in both males and females Dthe two sample means might be estimating the same true mean breath-holding time Etwo samples have the same variance (39% of 98 students) Null Hypothesis Males and females are NOT different so both samples estimate the same statistic : mean breath-holding time of humans

29 Q16The t-Test Two-Sample assuming Equal Variances test is used here to test whether: Aone-tail tests or two-tail tests are better to use in your hypothesis Bthe two samples have significantly different sample sizes Cbreath-holding can be held for significantly longer than zero seconds in both males and females Dthe two sample means might be estimating the same true mean breath-holding time Etwo samples have the same variance You cannot decide whether to use 1- or 2-tailed tests by looking for the best answer after in the results. The choice MUST made beforehand (a priori) independent of actual results Usually in biological sciences, use a 2-tailed hypothesis for t tests In two-sample t tests, the two samples are independent of each other, and do NOT have to have the same number of observations (unlike the paired t test) The N.H. is that the mean time is the same, not that it is zero! (logically it follows that the Hypothesized mean difference between the sample means should ideally be zero) NO! It is NOT testing whether variances are the same, it is ASSUMING that there are the same. t tests always test MEANS

30 Q18Which of the following statements is TRUE about t tests of the Type shown in Table 4 ? Athe variances are not identical, so it would have been better to use a t test assuming unequal variances Ba negative sign for a t value makes it less likely that a significant difference is present Cthe smaller the t value is, the less likely that the samples are different Dthe degrees of freedom are not influenced by sample size Ea zero in the Hypothesized Mean Difference box means that the t Test has proved that the samples are similar

31 Q18 A the variances are not identical, so it would have been better to use a t test assuming unequal variances FemalesMales Mean Variance Sample variances need not be identical only if one is about 3 times greater than the other, find out if significant with F test (Variance Ratio) So here it is OK to use the equal variance t test

32 Q18 Ba negative sign for a t value makes it less likely that a significant difference is present FemalesMales Mean t Stat The t Stat has a minus sign only because EXCEL subtracts the second column from the first = = tailed t tests IGNORE the sign test for a change in EITHER direction

33 Q18 Cthe smaller the t value is, the less likely that the samples are different N.H. Hypothesized Difference between Means = 0 Expected difference = 0 t Stat = (Observed difference) - (Expected difference) S.E. of the difference the smaller the t value is, the MORE likely that the samples are the SAME (SAMPLED FROM THE SAME POPULATION)

34 EXCEL t TESTS t Stat P value Hypothesis= NULL HYPOTHESIS (N.H.) Hypothesized Mean Difference = 0 The smaller the t value, the greater the agreement with the N.H. the less significant the difference from N.H. & so the greater the P value

35 Q18 Dthe degrees of freedom are not influenced by sample size For 2-sample t tests (independent sample t tests) df = (N 1 -1) + (N 2 - 2) = N 1 + N FemalesMales Mean Observations813 df19

36 Q18 Ea zero in the Hypothesized Mean Difference box means that the t Test has proved that the samples are similar FemalesMales Mean Hypothesized Mean Difference0 The hypothesis is set up before the test It is the t Stat that proves similarity if P value >0.05 that proves difference if P < 0.05 …but always a chance of being wrong!

37 EXCEL t TESTS t Stat P value Hypothesis= NULL HYPOTHESIS (N.H.) Hypothesized Mean Difference = 0 The bigger the t value, the greater the mismatch with the N.H. the more significant the difference from N.H. & the smaller the P value

38 Q21In a study on on touch discrimination, 12 male and 18 female subjects recorded the two-point discrimination distance on the upper surface of the hand. The most appropriate test to decide whether there was any difference in touch discrimination between the sexes at the upper surface of the hand (mean 7.0 +/- 4 mm S.D. for males, and 7.4 +/- 2 mm S.D. for females) is : ? alteration

39 Aa 2-tailed two-sample t test assuming equal variances Ba 2-tailed paired sample t test Ca 2-tailed two-sample t test assuming unequal variances Da 1-tailed paired sample t test Ea 1-tailed two-sample test assuming equal variances

40 The most appropriate test to decide whether there was any difference in touch discrimination between the sexes at the upper surface of the hand (mean 7 +/- 4mm S.D. for males, and 7.4 +/- 2mm S.D. for females) VARIANCE = SD 2 =? VARIANCE = SD 2 =? = 16 = 4 Variance ratio (F) = 16 / 4 = 4 so advise unequal variance t test AFTER confirmation by F test if P<0.05

41 Aa 2-tailed two-sample t test assuming equal variances Ba 2-tailed paired sample t test Ca 2-tailed two-sample t test assuming unequal variances Da 1-tailed paired sample t test Ea 1-tailed two-sample test assuming equal variances

42 OUTCOMES : you should be better able to distinguish between correct and incorrect statements about EXCEL t TESTS Lecture 2 : EXCEL t TESTS INTERPRETATION when COMPARING SAMPLES

43 EXCEL t TESTS t Stat P value Hypothesis= NULL HYPOTHESIS (N.H.) Hypothesized Mean Difference = 0 The bigger the t value, the greater the mismatch with the N.H. The smaller the P value, the more significant the difference from N.H. P is the PROBABILITY OF BEING THE SAME

44 EXCEL t TESTS What type ? Paired sample or unpaired 2-sample ? One-tailed or two-tailed ? Assuming Equal or Unequal Variances ? 2 sets of data on the same subjects? Or Data on 2 sets of subjects ? Use Equal unless variance of one sample more than about 3x the other Use two-tailed P values

45 revise.. MP110 WebCT SELF-TEST MCQ (Q11-Q22) on your EXCEL print-outs


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