1. Kin 304 Inferential Statistics
“Statistics means never having to say you're certain”

2. Inferential Statistics
As the name suggests Inferential Statistics allow us to make inferences about the population, based upon the sample, with a specified degree of confidence
Inferential Statistics

3. Inferential Statistics
The Scientific Method
Select a sample representative of the population. The method of sample selection is crucial to this process along with the sample size being large enough to allow appropriate probability testing.
Calculate the appropriate test statistic. The test statistic used is determined by the hypothesis being tested and the research design as a whole.
Test the Null hypothesis. Compare the calculated test statistic to its critical value at the predetermined level of acceptance.
Inferential Statistics

4. Setting a Probability Level for Acceptance
Prior to analysis the researcher must decide upon their level of acceptance.
Tests of significance are conducted at pre-selected probability levels, symbolized by p or α.
The vast majority of the time the probability level of 0.05, is used.
A p of .05 means that if you reject the null hypothesis, then you expect to find a result of this magnitude by chance only 5 in 100 times. Or conversely, if you carried out the experiment 100 times you would expect to find a result of this magnitude 95 times. You therefore have 95% confidence in your result. A more stringent test would be one where the p = 0.01, which translates to 99% confidence in the result.
Inferential Statistics

5. Inferential Statistics
No Rubber Yard Sticks
Either the researcher should pre-select one level of acceptance and stick to it, or do away with a set level of acceptance all together and simply report the exact probability of each test statistic.
If for instance, you had calculated a t statistic and it had an associated probability of p = 0.032, you could either say the probability is lower than the pre-set acceptance level of 0.05 therefore a significant difference at the 95% level of confidence or simply talk about as a percentage confidence (96.8%)
Inferential Statistics

6. Significance of Statistical Tests
The test statistic is calculated
The critical value of the test statistic is determined
based upon sample size and probability acceptance level (found in a table at the back of a stats book or part of the EXCEL stats report, or SPSS output)
The calculated test statistics must be greater than the critical value of the test statistic to accept a significant difference or relationship
Inferential Statistics

7. Degrees
Probability
of Freedom
0.05
0.01 1
.997
1.000 24
.388
.496 2
.950
.990 25
.381
.487 3
.878
.959 26
.374
.478 4
.811
.917 27
.367
.470 5
.754
.874 28
.361
.463 6
.707
.834 29
.355
.456 7
.666
.798 30
.349
.449 8
.632
.765 35
.325
.418 9
.602
.735 40
.304
.393 10
.576
.708 45
.288
.372 11
.553
.684 50
.273
.354 12
.532
.661 60
.250 13
.514
.641 70
.232
.302 14
.497
.623 80
.217
.283 15
.482
.606 90
.205
.267 16
.468
.590
100
.195
.254 17
.575
125
.174
.228 18
.444
.561
150
.159
.208 19
.433
.549
200
.138
.181 20
.423
.537
300
.113
.148 21
.413
.526
400
.098
.128 22
.404
.515
500
.088
.115 23
.396
.505
1,000
.062
.081
Table 2-4.2: Critical Values of the Correlation Coefficient

8. Kin 304 Tests of Differences between Means: t-tests
SEM
Visual test of differences
Independent t-test
Paired t-test

9. Comparison Is there a difference between two or more groups?
Test of difference between means
t-test
(only two means, small samples)
ANOVA - Analysis of Variance
Multiple means
ANCOVA
covariates
t Tests

10. Standard Error of the Mean
Describes how confident you are that the mean of the sample is the mean of the population
t Tests

11. Visual Test of Significant Difference between Means
1 Standard Error of the Mean
1 Standard Error of the Mean
Overlapping standard error bars therefore no significant difference between means of A and B A B
Mean
No overlap of standard error bars therefore a significant difference between means of A and B at about 95% confidence

12. Independent t-test
Two independent groups compared using an independent T-Test (assuming equal variances)
e.g. Height difference between men and women
The t statistic is calculated using the difference between the means in relation to the variance in the two samples
A critical value of the t statistic is based upon sample size and probability acceptance level (found in a table at the back of a stats book or part of the EXCEL t-test report, or SPSS output)
the calculated t based upon your data must be greater than the critical value of t to accept a significant difference between means at the chosen level of probability
t Tests

13. t statistic quantifies the degree of overlap of the distributions
t Tests

14. standard error of the difference between means
The variance of the difference between means is the sum of the two squared standard deviations.
The standard error (S.E.) is then estimated by adding the squares of the standard deviations, dividing by the sample size and taking the square root.
t Tests

15. t statistic
The t statistic is then calculated as the ratio of the difference between sample means to the standard error of the difference, with the degrees of freedom being equal to n - 2.
t Tests

16. Critical values of t Hypothesis: Degrees of Freedom = 2n – 2
There is a difference between means
Degrees of Freedom = 2n – 2
tcalc > tcrit = significant difference
t Tests

17. Paired Comparison Paired t Test
sometimes called t-test for correlated data
“Before and After” Experiments
Bilateral Symmetry
Matched-pairs data
t Tests

18. Paired t-test Hypothesis:
Is the mean of the differences between paired observations significantly different than zero
the calculated t statistic is evaluated in the same way as the independent test
t Tests

19. 9 Subjects All Lose Weight
Paired Weight Loss Data
n = 9
Weight Before (kg)
Weight After (kg)
Weight Loss (kg)
89.0
87.5
1.5
67.0
65.8
1.2
112.0
111.0
1.0
109.0
108.5
0.5
56.0
55.5
123.5
122.0
108.0
106.5
73.0
72.5
83.0
81.0
2.0
Mean of differences = +1.13

20. MS EXCEL t-Test: Independent WRONG ANALYSIS Before After Mean
Variance
531.11
Observations 9
Pooled Variance
Hypothesized Mean Difference df 16
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail

21. MS EXCEL t-Test: Paired CORRECT ANALYSIS Before After Mean 91.16666667
Variance
531.11
Observations 9
Pearson Correlation
Hypothesized Mean Difference df 8
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail

22. Kin 304 Tests of Differences between Means: ANOVA – Analysis of Variance
One-way ANOVA

23. ANOVA – Analysis of Variance
Used for analysis of multiple group means
Similar to independent t-test, in that the difference between means is evaluated based upon the variance about the means.
However multiple t-tests result in an increased chance of type 1 error.
F (ratio) statistic is calculated and is evaluated in comparison to the critical value of F (ratio) statistic
Tests of Difference – ANOVA

24. Tests of Difference – ANOVA
One-way ANOVA
One grouping factor
HO: The population means are equal
HA: At least one group mean is different
Two or more levels of grouping factor
Exposure = low, medium or high
Age Groups = 7-8, 9-10, 11-12, 13-14
Tests of Difference – ANOVA

25. Tests of Difference – ANOVA
F (ratio) Statistic
The F ratio compares two sources of variability in the scores.
The variability among the sample means, called Between Group Variance, is compared with the variability among individual scores within each of the samples, called Within Group Variance.
Tests of Difference – ANOVA

26. Formula for sources of variation
Tests of Difference – ANOVA

27. Anova Summary Table SS df MS F Between Groups SS(Between) k-1
MS(Between) MS(Within)
Within Groups
SS(Within)
N-k
SS(Within) N-k
Total
SS(Within) + SS(Between)
N-1 .
Tests of Difference – ANOVA

28. Tests of Difference – ANOVA
Assumptions for ANOVA
The populations from which the samples were obtained are approximately normally distributed.
The samples are independent.
The population value for the standard deviation between individuals is the same in each group.
If standard deviations are unequal transformation of values may be needed.
Tests of Difference – ANOVA

29. CFS Kids 17 – 19 years (Boys) ANOVA Dependent - VO2max
Grouping Factor - Age (17, 18, 19)
No Significant difference between means for VO2max (p>0.05)

30. CFS Kids 17 – 19 years (Girls)
ANOVA
Dependent - VO2max
Grouping Factor - Age (17, 18, 19)
Significant difference between means for VO2max (p<0.05)

31. Tests of Difference – ANOVA
Post Hoc tests
Post hoc simply means that the test is a follow-up test done after the original ANOVA is found to be significant.
One can do a series of comparisons, one for each two-way comparison of interest.
E.g. Scheffe or Tukey’s tests
The Scheffe test is very conservative
Tests of Difference – ANOVA

32. Scheffe’s – Post Hoc Test
Boys
Scheffe’s – Post Hoc Test
Girls
Boys – no significant differences, would not run post hoc tests
Girls – VO2max for age19 is significantly different than at age17

33. ANOVA – Factorial design Multiple factors
Test of differences between means with two or more grouping factors, such that each factor is adjusted for the effect of the other
Can evaluate significance of factor effects and interactions between them
2 – way ANOVA: Two factors considered simultaneously
Tests of Difference – ANOVA

34. Significant difference in VO2max (p<0.05) by SEX=Main effect
Example: 2 way ANOVA
Dependent - VO2max
Grouping Factors
AGE (17, 18, 19)
SEX (1, 2)
Significant difference in VO2max (p<0.05) by SEX=Main effect
Significant difference in VO2max (p<0.05) by AGE=Main effect
No Significant Interaction (p<0.05) AGE * SEX

35. Analysis of Covariance (ANCOVA)
Taking into account a relationship of the dependent with another continuous variable (covariate) in testing the difference between means of one or more factor
Tests significance of difference between regression lines
Tests of Difference – ANOVA

36. Scatterplot showing correlations between skinfold-adjusted Forearm girth and maximum grip strength for men and women

37. Use of T tests for difference between means?
Men are significantly (p<0.05) bigger than women in skinfold-adjusted forearm girth and grip strength

38. ANCOVA Dependent – Maximum Grip Strength (GRIPR) Grouping Factor – Sex Covariate – Skinfold-adjusted Forearm Girth (SAFAGR)
SAFAGR is a significant Covariate
No significant difference between sexes in Grip Strength when adjusted for Covariate (representing muscle size)
Therefore one regression line (not two, for each sex) fit the relationship

39. Tests of Difference – ANOVA
3-way ANOVA
For 3-way ANOVA, there will be:
- three 2-way interactions (AxB, AxC) (BxC)
- one 3-way interaction (AxBxC)
If for each interaction (p > 0.05) then use main effects results
Typically ANOVA is used only for 3 or less grouping factors
Tests of Difference – ANOVA

40. Repeated Measures ANOVA
Repeated measures design – the same variable is measured several times over a period of time for each subject
Pre- and post-test scores are the simplest design – use paired t-test
Advantage - using fewer experimental units (subjects) and providing a control for differences (effect of variability due to differences between subjects can be eliminated)
Tests of Difference – ANOVA