Kin 304 Inferential Statistics

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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
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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

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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.

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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.

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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)

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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

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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
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Kin 304Tests of Differences between Means
t-tests

• SEM
• Visual test of differences
• Independent t-test
• Paired t-test

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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

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

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t statistic quantifiesthe degree of overlap of
the distributions

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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.

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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.

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Critical values of t

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

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Paired Comparison

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

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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

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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
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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
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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
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Kin 304Tests of Differences between MeansANOVA
Analysis of Variance

• One-way ANOVA

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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

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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

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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.

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Formula for sources of variation
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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 .
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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.

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CFS Kids 17 19 years (Boys)

• ANOVA
• Dependent - VO2max
• Grouping Factor - Age (17, 18, 19)
• No Significant difference between means for
  VO2max (pgt0.05)

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CFS Kids 17 19 years (Girls)

• ANOVA
• Dependent - VO2max
• Grouping Factor - Age (17, 18, 19)
• Significant difference between means for VO2max
  (plt0.05)

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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

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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

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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

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• 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

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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

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

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Use of T tests for difference between means?

• Men are significantly (plt0.05) bigger than women
  in skinfold-adjusted forearm girth and grip
  strength

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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

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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

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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)