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A CAL COMPANION TO EXCEL PRINT-OUTS

MP110 SKILLS FOR LIFE SCIENCESLecture 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.

MP110TWO REVISION LECTURES ON STATISTICS

Dr Mike Jakobson

Lecture 1 :INITIAL EXCEL DATA DESCRIPTION before COMPARING SAMPLES

Lecture 2 :EXCEL t TESTS INTERPRETATIONwhen COMPARING SAMPLES

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 & DysfunctionPP250 Physiological Regulation

OUTCOMES : you should be better able to

distinguish between correct and incorrect statements

about EXCEL t TESTS

Lecture 2 :EXCEL t TESTS INTERPRETATIONwhen COMPARING SAMPLES

HANDOUT in YOUR MP110 WORKBOOK

Also files on INTRANET

DEPARTMENT MENUDirectory : Skills for Life SciencesMPA100 Semester B data handling

MPA110_MJSTATS FILE ONE_QUESTIONS.docMPA110_MJSTATS FILE TWO_ANSWERS.doc

MP110_MJSTATS LECTURE 1.ppsMP110_MJSTATS LECTURE 2.pps

also see..MP110 WebCT

SELF-TEST MCQ(Q11-Q22)

on yourEXCEL print-outs

EXCEL t TESTS

t Stat

P value

Hypothesis = NULL HYPOTHESIS (N.H.) “Hypothesized Mean Difference” = 0

The bigger the t value,the greater the mismatchwith the N.H.

The smaller the P value,the more significant the difference from N.H.

EXCEL t TESTS

What type of t test ?

Paired sample or unpaired 2-sample ?

One-tailed or two-tailed ?

Assuming Equal or Unequal Variances ?

Q20 In 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

Q20 In 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 forMEAN CHANGE IN INDIVIDUAL B.P.

Interested in change in EITHER direction (up or down)...................so use

a TWO-TAILED t test

Standing Supine Change116 123 -7130 119 11118 120 -2122 113 9126 114 12127 117 10124 118 6125 123 2

124 118 65 4 7

MEANS.D.

The paired t testlooks at the

8 values for CHANGE

N.H. could the MEAN

be estimatinga ‘zero change’?

Table 3t-Test: Paired Two Sample for Means

a) NORM f) DB:AIRMean 30.83333 40.27778Variance 99.55882 193.0359Observations 18 18Pearson Correlation 0.282101

Hypothesized Mean Difference 0df 17t Stat -2.73666P(T<=t) one-tail 0.007028

t Critical one-tail 1.739606P(T<=t) two-tail 0.014055

t Critical two-tail 2.109819

df NOT Df

t not Tlower case capital

For Q13 , Q14, Q15

Q13 In 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’

Q13 In 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 samplednot influenced by

those already sampled

POPULATIONbeing sampled

Supposea high valueis randomly selectedThen, 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”

t-Test: Paired Two Sample for Means

NORM DB:AIRMean 30.83333 40.27778

Q13 In 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 MISGUIDEDnot PAIRED SAMPLE FOR MEANS,

DATA VALUES are paired offnot the MEANS.

MEAN INDIVIDUAL CHANGE

t-Test: Paired Two Sample for Means

NORM DB:AIRMean 30.83333 40.27778

Just one MEANis being tested,

namely the MEAN

INDIVIDUAL CHANGE

NORM - DB:AIR

= - 18.41435

MEAN INDIVIDUAL CHANGE

Q13 In 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

t-Test: Paired Two Sample for Means

a) NORM f) DB:AIRMean 30.83333 40.27778Variance 99.55882 193.0359Observations 18 18Pearson Correlation 0.282101

Hypothesized Mean Difference 0df 17t Stat -2.73666P(T<=t) one-tail 0.007028

t Critical one-tail 1.739606P(T<=t) two-tail 0.014055

t Critical two-tail 2.109819

Table 3df = degrees of freedom = (N - 1)

N = number of values used in t test

PAIRED t test USES NUMBER OF PAIRSso N=18 (not 36), and df = 18-1 = 17

the ‘value’ used is the differencebetween each pair of observations

Q13 In 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 = S2

=S.D.

NS.E.M.

= S.D.NS.E.M. x

(S.E.M.)2 x N = Variance

t-Test: Paired Two Sample for Means

a) NORM f) DB:AIRMean 30.83333 40.27778Variance 99.55882 193.0359Observations 18 18Pearson Correlation 0.282101

Hypothesized Mean Difference 0df 17t Stat -2.73666P(T<=t) one-tail 0.007028

t Critical one-tail 1.739606P(T<=t) two-tail 0.014055

t Critical two-tail 2.109819

Table 3

Variances are given in the print-outBUT

in a PAIRED t TESTit does not alter validity of t value

if they are different

though it is still physiologically relevant

Q13 In 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 fromthe population being studied

NO! The whole idea is to look at change

in performance WITHIN an individual

Q14 In Table 3, the t value (correct to 3 decimal places) calculated by the t test from the breath-holding data is :

A -2.736.B 2.736C 1.740D 2.110E -2.737

Correctly roundedand without

confusing extra ‘.’

Q15 Assuming 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 lowercan be regarded as

‘significant’

So long as itis UNLIKELY tohave occurredby chance

i.e. P less than (<)5% (P<0.05)

Q15 Assuming 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 -2.73666 P(T<=t) two-tail 0.014055 t Critical two-tail 2.109819

P = probability that

Null Hypothesis

(of NO change)

is TRUEt Stat =

t value calculated from the OBSERVED

data

t Critical

= largest t value (+ or -)

EXPECTED

from Null Hypothesis

at = 0.05 (5% probability)

Q15 Assuming 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 -2.73666 P(T<=t) two-tail 0.014055 t Critical two-tail 2.109819

49% studentsgot this right

Not a P value

not <0.01, only <0.05

exact P values OK!

wrong way round!

is P OK?

t-Test: Two-Sample Assuming Equal VariancesDB:AIR DB:AIR

Females MalesMean 31.625 42.15385 Variance 243.9821 137.4744 Observations 8 13 Pooled Variance 176.7141 Hypothesized Mean Difference 0 df 19 t Stat -1.76259 P(T<=t) one-tail 0.047024 t Critical one-tail 1.729131 (T<=t) two-tail 0.094048 t Critical two-tail 2.093025

Table 4 Q16 and Q18

Q16 The t-Test Two-Sample assuming Equal Variances test is used here to test whether:

A one-tail tests or two-tail tests are better to use in your hypothesis

B the two samples have significantly different sample sizes

C breath-holding can be held for significantly longer than zero seconds in both males and females

D the two sample means might be estimating the same true mean breath-holding time

E two samples have the same variance

(39% of ’98 students)

Null HypothesisMales and females are NOT different

so both samples estimate thesame statistic :

mean breath-holding time of humans

Q16 The t-Test Two-Sample assuming Equal Variances test is used here to test whether:

A one-tail tests or two-tail tests are better to use in your hypothesis

B the two samples have significantly different sample sizes

C breath-holding can be held for significantly longer than zero seconds in both males and females

D the two sample means might be estimating the same true mean breath-holding time

E two samples have the same variance

You cannot decide whether to use1- 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 samplesare independent of each other, anddo NOT have to have the same numberof 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 betweenthe 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

Q18 Which of the following statements is TRUE about t tests of the Type shown in Table 4 ?

A the variances are not identical, so it would have been better to use a t test ‘assuming unequal variances’

B a negative sign for a t value makes it less likely that a significant difference is present

C the smaller the t value is, the less likely that the samples are different

D the degrees of freedom are not influenced by sample size

E a zero in the Hypothesized Mean Difference box means that the t Test has proved that the samples are similar

Q18A the variances are not identical, so it would have been better to use a t test ‘assuming unequal variances’

Females MalesMean 31.625 42.15385 Variance 243.9821 137.4744

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

Q18B a negative sign for a t value makes it less likely

that a significant difference is present

Females MalesMean 31.625 42.15385 t Stat -1.76259

The t Stat has a minus sign only becauseEXCEL subtracts the second column from the first

= 31.625 - 42.15385= - 1.76259

2-tailed t tests IGNORE the sign test for a change in EITHER direction

Q18C the 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)

EXCEL t TESTS

t Stat

P value

Hypothesis = NULL HYPOTHESIS (N.H.) “Hypothesized Mean Difference” = 0

The smaller the t value,the greater the agreementwith the N.H.

the less significant the difference from N.H.& so the greater the P value

Q18D the degrees of freedom are not influenced by sample size

For 2-sample t tests (independent sample t tests)df = (N1 -1) + (N2 - 2) = N1 + N2 - 2

Females MalesMean 31.625 42.15385 Observations 8 13 df 19

Q18E a zero in the Hypothesized Mean Difference box

means that the t Test has proved that the samples are similar

Females MalesMean 31.625 42.15385

Hypothesized Mean Difference 0

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!

EXCEL t TESTS

t Stat

P value

Hypothesis = NULL HYPOTHESIS (N.H.) “Hypothesized Mean Difference” = 0

The bigger the t value,the greater the mismatchwith the N.H.

the more significantthe difference from N.H.& the smaller the P value

Q21 In 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

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

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 = SD2

=?

VARIANCE = SD2

=?= 16

= 4

Variance ratio (F) = 16 / 4 = 4so advise unequal variance t test

AFTER confirmation by F test if P<0.05

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

OUTCOMES : you should be better able to

distinguish between correct and incorrect statements

about EXCEL t TESTS

Lecture 2 :EXCEL t TESTS INTERPRETATIONwhen COMPARING SAMPLES

EXCEL t TESTS

t Stat

P value

Hypothesis = NULL HYPOTHESIS (N.H.) “Hypothesized Mean Difference” = 0

The bigger the t value,the greater the mismatchwith the N.H.

The smaller the P value,the more significant the difference from N.H.

P is the PROBABILITY

OF BEING THE SAME

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 onesample more than about 3x the other

Use two-tailed P values

revise..MP110 WebCT

SELF-TEST MCQ(Q11-Q22)

on yourEXCEL print-outs