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Kin 304Inferential Statistics

- Probability Level for Acceptance
- Type I and II Errors
- One and Two-Tailed tests
- Critical value of the test statistic

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

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.

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 a. - 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.

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)

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

Degrees Probability Probability Degrees Probability Probability

of Freedom 0.05 0.01 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 .325

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 .456 .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 Table 2-4.2 Critical Values of the Correlation Coefficient Table 2-4.2 Critical Values of the Correlation Coefficient Table 2-4.2 Critical Values of the Correlation Coefficient Table 2-4.2 Critical Values of the Correlation Coefficient Table 2-4.2 Critical Values of the Correlation Coefficient Table 2-4.2 Critical Values of the Correlation Coefficient

Kin 304Tests 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

Standard Error of the Mean

Describes how confident you are that the mean of

the sample is the mean of the population

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 statistic quantifiesthe degree of overlap of

the distributions

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 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.

Critical values of t

- Hypothesis
- There is a difference between means
- Degrees of Freedom 2n 2
- tcalc gt tcrit significant difference

Paired Comparison

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

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

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 0.5

123.5 122.0 1.5

108.0 106.5 1.5

73.0 72.5 0.5

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 9

Pooled Variance 534.4925

Hypothesized Mean Difference 0

df 16

t Stat 0.103990367

P(Tltt) one-tail 0.459234679

t Critical one-tail 1.745884219

P(Tltt) 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 9

Pearson Correlation 0.999741718

Hypothesized Mean Difference 0

df 8

t Stat 6.23354978

P(Tltt) one-tail 0.000125066

t Critical one-tail 1.85954832

P(Tltt) two-tail 0.000250133

t Critical two-tail 2.306005626

Kin 304Tests of Differences between MeansANOVA

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

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

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.

Formula for sources of variation

Anova Summary Table

SS df MS F

Between Groups SS(Between) k-1 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 .

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.

CFS Kids 17 19 years (Boys)

- ANOVA
- Dependent - VO2max
- Grouping Factor - Age (17, 18, 19)
- No Significant difference between means for

VO2max (pgt0.05)

CFS Kids 17 19 years (Girls)

- ANOVA
- Dependent - VO2max
- Grouping Factor - Age (17, 18, 19)
- Significant difference between means for VO2max

(plt0.05)

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 Tukeys tests
- The Scheffe test is very conservative

Scheffes Post Hoc Test

Boys

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

- Example 2 way ANOVA
- Dependent - VO2max
- Grouping Factors
- AGE (17, 18, 19)
- SEX (1, 2)
- Significant difference in VO2max (plt0.05) by

SEXMain effect - Significant difference in VO2max (plt0.05) by

AGEMain effect - No Significant Interaction (plt0.05) 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

- 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 (plt0.05) bigger than women

in skinfold-adjusted forearm girth and grip

strength

ANCOVADependent 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

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 gt 0.05) then use main

effects results - Typically ANOVA is used only for 3 or less

grouping factors

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)