Hypothesis Testing Introduction to Study Skills & Research Methods (HL10040) Dr James Betts.

Hypothesis Testing

Introduction to Study Skills & Research Methods (HL10040)

Dr James Betts

Lecture Outline:

•What is Hypothesis Testing?

•Hypothesis Formulation

•Statistical Errors

•Effect of Study Design

•Test Procedures

•Test Selection.

Statistics

Descriptive Inferential

Correlational

Relationships

GeneralisingOrganising, summarising & describing data

Significance

What is Hypothesis Testing?

A B A B

Null Hypothesis

We also need to establish:

1) How …………………….. are these observations?

2) Are these observations reflective of the ………………………….?

Alternative Hypothesis

Example Hypotheses: Isometric Torque• Is there any difference in the length of time that males and

females can sustain an isometric muscular contraction?

Null Hypothesis

There is not a significant difference in the DV between males and females

Alternative Hypothesis

There is a significant difference in the DV between males and females.

Example Hypotheses: Isometric Torque• Is there any difference in the length of time that males and

females can sustain an isometric muscular contraction?

Energy Intake (calories per day)

1500 2500 3500 4500 5500

Nu

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16 17 18 19 20

Sustained Isometric Torque (seconds)

N♂N♀

n♂n♀

Statistical Errors• Type 1 Errors

-Rejecting H0 when it is actually true -Concluding a difference when one does not actually exist

• Type 2 Errors

-Accepting H0 when it is actually false (e.g. previous slide)-Concluding no difference when one does exist

Energy Intake (calories per day)

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Sustained Isometric Torque (seconds)

n♂n♀

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Step 1:

Calculate the Standard Error for Each Mean

SEM♀ = SD/√n =

SEM♂ = SD/√n =

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Step 2:

Calculate the Standard Error for the difference in means

SEMdiff = √ SEM♀2 + SEM♂2 =

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Step 3:

Calculate the t statistic

t = (Mean♀ - Mean♂) / SEMdiff =

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Step 4:

Calculate the degrees of freedom (df)

df = (n♀ - 1) + (n♂ - 1) =

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Step 5:

Determine the critical value for t using a t-distribution table

Degrees of Freedom Critical t-ratio

44464850

2.0152.0132.0112.009

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Step 6 finished:

Compare t calculated with t critical

Calculated t =

Critical t =

Independent t-test: Calculation

Mean SD n

♀ 18.5 1.74 25

♂ 17.5 1.72 25

Evaluation:

The wealth of available literature supports that females can sustain isometric contractions longer than males. This may suggest that the findings of the present study represent a type error

Possible solution:

Independent t-test: SPSS Output

Independent Samples Test

7.842 .012 -2.333 18 .031 -1.69600 .72710 -3.22358 -.16842

-2.333 15.447 .034 -1.69600 .72710 -3.24188 -.15012

Equal variancesassumed

Equal variancesnot assumed

SwimTime50mF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

Group Statistics

10 24.7720 1.25246 .39606

10 26.4680 1.92823 .60976

GroupControl

Visualisation

SwimTime50mN Mean Std. Deviation

Std. ErrorMean

Swim Data from SPSS session 8

Week1 2

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-Up

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180

200

Advantages of using Paired Data• Data from independent samples is heavily

influenced by variance between subjects

Week1 2

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180

200 Paired t-test: Calculation

Mean SD n

Week 1 61.6 56.6 8

Week 2 65.5 57.5 8

…a paired t-test can use the specific differences between each pair rather than just subtracting

mean A from mean B

(see earlier step 3)

Paired t-test: CalculationSubject Week 1 Week 2 Diff (D) Diff2 (D2)

1 10 12

2 50 52

3 20 25

4 8 10

5 115 120

6 75 80

7 45 50

8 170 175∑D = ∑D2 =Steps 1 & 2: Complete this table

Paired t-test: Calculation

Step 3:

Calculate the t statistic

t = n x ∑D2 – (∑D)2 = √ (n - 1)

∑D

Paired t-test: Calculation

Steps 4 & 5:

Calculate the df and use a t-distribution table to find t critical

Degrees of FreedomCritical t-ratio

(0.05 level)

123456789

12.714.3033.1822.7762.5712.4472.3652.3062.262

Critical t-ratio (0.01

level)63.6579.9255.8414.6044.0323.7073.4993.3553.250

Paired t-test: CalculationStep 6 finished:

Compare t calculated with t critical

Calculated t =

Critical t =

Mean SD n

Week 1 61.6 56.6 8

Week 2 65.5 57.5 8

Paired Samples Test

-3.87500 1.55265 .54894 -5.17305 -2.57695 -7.059 7 .000VAR00001 - VAR00002Pair 1Mean Std. Deviation

Std. ErrorMean Lower Upper

95% ConfidenceInterval of the

Difference

Paired Differences

t df Sig. (2-tailed)

Paired t-test: SPSS Output

Push-up Data from lecture 3

Paired Samples Statistics

61.6250 8 56.64157 20.02582

65.5000 8 57.54005 20.34348

VAR00001

VAR00002

Pair1

Mean N Std. DeviationStd. Error

Mean

Example Hypotheses: Isometric Torque• Is there any difference in the length of time that males and

females can sustain an isometric muscular contraction?

Energy Intake (calories per day)

1500 2500 3500 4500 5500

Nu

mb

er o

f P

eo

ple

0

20

40

60

80

100

120

140

160

16 17 18 19 20

Sustained Isometric Torque (seconds)

t-test

Mean A

Mean B

Example Hypotheses: Isometric Torque• Is there any difference in the length of time that males and

females can sustain an isometric muscular contraction?

Energy Intake (calories per day)

1500 2500 3500 4500 5500

Nu

mb

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f P

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ple

0

20

40

60

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100

120

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16 17 18 19 20

Sustained Isometric Torque (seconds)

Mean A

Mean B

…assumptions of parametric analyses

• All data or paired differences are ND (this is the main consideration)

• N acquired through random sampling

• Data must be of at least the interval LOM

• Data must be Continuous.

Non-Parametric Tests

• These tests use the median and do not assume anything about distribution, i.e. ‘distribution free’

• Mathematically, value is ignored (i.e. the magnitude of differences are not compared)

• Instead, data is analysed simply according to rank.

Non-Parametric Tests

• Independent Measures

– Mann-Whitney Test

• Repeated Measures

– Wilcoxon Test

Mann-Whitney U: CalculationStep 1:

Rank all the data from both groups in one series, then total each

Student

School A School B

StudentGrade GradeRank Rank

J. S. L. D. H. L. M. J. T. M. T. S. P. H.

T. J. M. M. K. S. P. S. R. M. P. W. A. F.

B- B- A+ D- B+ A- F

D C+ C+ B- E C-

A-

9 9

14 3

11 12.5 1

4 6.5 6.5 9 2 5

12.5

Median = ; Median = ;∑RA = ∑RB =

Mann-Whitney U: CalculationStep 2:

Calculate two versions of the U statistic using:

U1 = (nA x nB) + 2

(nA + 1) x nA - ∑RA

AND…

U2 = (nA x nB) + 2

(nB + 1) x nB - ∑RB

Mann-Whitney U: CalculationStep 3 finished:

Select the smaller of the two U statistics (U1 = ………; U2 = ……..)

…now consult a table of critical values for the Mann-Whitney test

n

0.05

0.01

6

5

2

7

8

4

8

13

7

9

17

11

Calculated U must be critical U to conclude a significant difference

Conclusion

Median A Median B

Test Statisticsb

17.500

45.500

-.900

.368

.383a

Mann-Whitney U

Wilcoxon W

Z

Asymp. Sig. (2-tailed)

Exact Sig. [2*(1-tailedSig.)]

VAR00001

Not corrected for ties.a.

Grouping Variable: VAR00002b.

Mann-Whitney U: SPSS OutputRanks

7 8.50 59.50

7 6.50 45.50

14

VAR000021.00

2.00

Total

VAR00001N Mean Rank Sum of Ranks

Wilcoxon Signed Ranks: CalculationStep 1:

Rank all the diffs from in one series (ignoring signs), then total each

AthletePre-training OBLA (kph)

Rank

J. S. L. D. H. L. M. J. T. M. T. S. P. H.

15.6 17.2 17.7 16.5 15.9 16.7

17.0

0.5 0.3 -1 0.3

0.1 -0.2 0.1 ∑Signed Ranks =

Post-training OBLA (kph)

Diff. Signed Ranks

16.1 17.5 16.7 16.8 16.0 16.5

17.1

6 4.5 -7 4.5 1.5 -3

1.5

- +

-7

-3

6 4.5

4.5

1.5

1.5

Medians =

Wilcoxon Signed Ranks: CalculationStep 2:

The smaller of the T values is our test statistic (T+ = ….....; T- = ……)

…now consult a table of critical values for the Wilcoxon test

n

0.05

6

0

7

2

8

3

9

5

Calculated T must be critical T to conclude a significant difference

Conclusion

Median A Median B

Test Statisticsb

-1.364a

.172

Z

Asymp. Sig. (2-tailed)

VAR00002 -VAR00001

Based on negative ranks.a.

Wilcoxon Signed Ranks Testb.

Wilcoxon Signed Ranks: SPSS OutputRanks

2a 3.00 6.00

5b 4.40 22.00

0c

7

Negative Ranks

Positive Ranks

Ties

Total

VAR00002 - VAR00001N Mean Rank Sum of Ranks

VAR00002 < VAR00001a.

VAR00002 > VAR00001b.

VAR00002 = VAR00001c.

So which stats test should you use?

Q1. What is the …………?

Q2. Is the data …….?

Q3. Is the data

……………..

or

……………..?