4 What is Hypothesis Testing? Null HypothesisAlternative HypothesisA BA BWe also need to establish:1) How …………………….. are these observations?Inevitably, especially with continuous data, score A and score B are never identical. So we need statistical procedures to assess ‘how’ different they are and whetehr we are happy to generalise this difference back to the larger (target) population from which we sampled.2) Are these observations reflective of the ………………………….?
5 Example Hypotheses: Isometric Torque Is there any difference in the length of time that males and females can sustain an isometric muscular contraction?Alternative HypothesisThere is a significant difference in the DV between males and females.Null HypothesisThere is not a significant difference in the DV between males and femalesWe start with what we think will be true and state this as a hypothesis, then we must state a null hypothesis. IMPROTANTLY, THIS MUST COVER ALL BASES. The result of the study must support one of these two hypotheses- the easiest way to do this is often just to insert the word not or no somewhere.PLEASE NOTE THAT THIS IS WHAT IS KNOWN AS A 2 TAILED OR NON-DIRECTIONAL HYPOTHESIS- i.e. we are just predicting a difference between groups, not that one is higher than the other.
6 Example Hypotheses: Isometric Torque Is there any difference in the length of time that males and females can sustain an isometric muscular contraction?N♀N♂n♀n♂POOR AND INSUFFICIENT ARE NOTED HERE. Clearly a random sample will increase the likelyhood that n will equal N. However, it would also follow that the larger n is the more accurate it will be…Sustained Isometric Torque (seconds)
7 Statistical Errors Type 1 Errors -Rejecting H0 when it is actually true-Concluding a difference when one does not actually existType 2 Errors-Accepting H0 when it is actually false (e.g. previous slide)-Concluding no difference when one does existNon-parametric tests are more likely to commit type 2 errors (i.e. less powerful so can miss true significant differences)
8 Independent t-test: Calculation Sustained Isometric Torque (seconds)MeanSDn♀18.51.7425♂17.51.72Now I know that many of you have been questioning the value of SPSS over excel but this manual calculation should demonstrate why SPSS can be so useful
9 Independent t-test: Calculation Step 1:Calculate the Standard Error for Each MeanSEM♀ = SD/√n =SEM♂ = SD/√n =MeanSDn♀18.51.7425♂17.51.72
10 Independent t-test: Calculation Step 2:Calculate the Standard Error for the difference in meansSEMdiff = √ SEM♀2 + SEM♂2 =MeanSDn♀18.51.7425♂17.51.72
11 Independent t-test: Calculation Step 3:Calculate the t statistict = (Mean♀ - Mean♂) / SEMdiff =MeanSDn♀18.51.7425♂17.51.72So basically recruiting more subjects and controlling the experiment will make you more likely to find a significant difference.VIP TO NOTE FOR LATER THAT BIG VARIANCE IS THEREFORE A BAD THINGBUT WE STILL NEED TO CONVERT OUR t VALUE INTO A P VALUE…
12 Independent t-test: Calculation Step 4:Calculate the degrees of freedom (df)df = (n♀ - 1) + (n♂ - 1) =MeanSDn♀18.51.7425♂17.51.72We calculate df quite simply as follows and then only use it to determine a critical value for t.
13 Independent t-test: Calculation Step 5:Determine the critical value for t using a t-distribution tableDegrees of FreedomCritical t-ratio444648502.0152.0132.0112.009MeanSDn♀18.51.7425♂17.51.72Note that you should always choose the column for 0.05 and for a 2 tailed test.Importantly, NOTE THE CONTINUED IMPORTANCE OF SAMPLE SIZE, THE MORE SUBJECTS WE HAVE THE LOWER THE CRITICAL VALUE GETS. THIS IS RELEVANT BECAUSE WE NEED OUR CALCULATED t TO EXCEED THE CRITICAL t TO CONCLUDE A STATISTICAL DIFFERENCE…
14 Independent t-test: Calculation Step 6 finished:Compare t calculated with t criticalCalculated t =Critical t =MeanSDn♀18.51.7425♂17.51.72
15 Independent t-test: Calculation 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 errorPossible solution:MeanSDn♀18.51.7425♂17.51.72
16 Independent t-test: SPSS Output Swim Data from SPSS session 8NOT THE SAME DATA AS WE JUST DID MANUALLY
17 Advantages of using Paired Data Data from independent samples is heavily influenced by variance between subjectsi.e. SD is large because one individual performed higher than another but in both trials!!!
18 Paired t-test: Calculation …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)MeanSDnWeek 161.656.68Week 265.557.5
20 Paired t-test: Calculation Step 3:Calculate the t statistict = n x ∑D2 – (∑D)2 = √ (n - 1)∑D
21 Paired t-test: Calculation Steps 4 & 5:Calculate the df and use a t-distribution table to find t criticalCritical t-ratio (0.05 level)Critical t-ratio (0.01 level)Degrees of Freedom12345678912.714.3033.1822.7762.5712.4472.3652.3062.26263.6579.9255.8414.6044.0323.7073.4993.3553.250Same table as before but now I am showing you an extra column
22 Paired t-test: Calculation Step 6 finished:Compare t calculated with t criticalCalculated t =Critical t =MeanSDnWeek 161.656.68Week 265.557.5
23 Paired t-test: SPSS Output Push-up Data from lecture 3
24 Example Hypotheses: Isometric Torque Is there any difference in the length of time that males and females can sustain an isometric muscular contraction?t-testMean AMean BSustained Isometric Torque (seconds)
25 Example Hypotheses: Isometric Torque Is there any difference in the length of time that males and females can sustain an isometric muscular contraction?Mean AMean BComparing the means does not give a valid reflection of the group differences.Sustained Isometric Torque (seconds)
26 …assumptions of parametric analyses All data or paired differences are ND (this is the main consideration)N acquired through random samplingData must be of at least the interval LOMData must be Continuous.
27 Non-Parametric TestsThese 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.This also now shows why ordinal data should be analysed using these tests.
28 Non-Parametric Tests Independent Measures Repeated Measures Mann-Whitney TestRepeated MeasuresWilcoxon TestThis also now shows why ordinal data should be analysed using these tests.
29 Mann-Whitney U: Calculation Step 1:Rank all the data from both groups in one series, then total eachSchool ASchool BStudentGradeRankStudentGradeRankJ. S. L. D. H. L. M. J. T. M. T. S. P. H.B- B- A+ D- B+ A- FT. J. M. M. K. S. P. S. R. M. P. W. A. F.D C+ C+ B- E C- A-Median = ;∑RA =Median = ;∑RB =
30 Mann-Whitney U: Calculation Step 2:Calculate two versions of the U statistic using:U1 = (nA x nB) +(nA + 1) x nA- ∑RA2AND…U2 = (nA x nB) +(nB + 1) x nB- ∑RB2
31 Mann-Whitney U: Calculation Step 3 finished:Select the smaller of the two U statistics (U1 = ………; U2 = ……..)…now consult a table of critical values for the Mann-Whitney testn0.050.01652784813791711ConclusionMedian A Median BCalculated U must be critical U to conclude a significant difference
33 Wilcoxon Signed Ranks: Calculation Step 1:Rank all the diffs from in one series (ignoring signs), then total eachPre-training OBLA (kph)Post-training OBLA (kph)AthleteDiff.RankSigned RanksJ. S. L. D. H. L. M. J. T. M. T. S. P. H.-7-3Medians =∑Signed Ranks =
34 Wilcoxon Signed Ranks: Calculation Step 2:The smaller of the T values is our test statistic (T+ = ….....; T- = ……)…now consult a table of critical values for the Wilcoxon testn0.056728395ConclusionMedian A Median BCalculated T must be critical T to conclude a significant difference