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

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

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

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

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 0.032 as a percentage confidence (96.8%) Inferential Statistics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

MS EXCEL t-Test: Independent WRONG ANALYSIS Before After Mean 91.16666667 90.03333333 Variance 537.875 531.11 Observations 9 Pooled Variance 534.4925 Hypothesized Mean Difference df 16 t Stat 0.103990367 P(T<=t) one-tail 0.459234679 t Critical one-tail 1.745884219 P(T<=t) two-tail 0.918469359 t Critical two-tail 2.119904821

MS EXCEL t-Test: Paired CORRECT ANALYSIS Before After Mean 91.16666667 90.03333333 Variance 537.875 531.11 Observations 9 Pearson Correlation 0.999741718 Hypothesized Mean Difference df 8 t Stat 6.23354978 P(T<=t) one-tail 0.000125066 t Critical one-tail 1.85954832 P(T<=t) two-tail 0.000250133 t Critical two-tail 2.306005626

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

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

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

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

Formula for sources of variation Tests of Difference – ANOVA

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

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

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)

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

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

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

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

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

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

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

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

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

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

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