This document provides information on various statistical concepts and tests. It defines descriptive statistics like mean, median, percentiles. It discusses hypothesis testing, types of errors, assumptions of normality, and transformations. It explains parametric vs non-parametric tests, ANOVA, correlations, regressions and their interpretations. Post-hoc tests for multiple comparisons are also summarized.

2. Descriptive statistics
Minimum
The smallest value.
25th percentile
25% of values are lower than this.
Median
Half the values are lower; half are higher.
75th percentile
75% of values are higher than this.
Maximum
The largest value.

3. Mean
The average.
Standard Deviation
Quantifies variability or scatter.
Standard Error of Mean
Quantifies how precisely the mean is known. It is a measure of how far your
sample mean is likely to be from the true population mean.
95% confidence interval
Given some assumptions, there is a 95% chance that this range includes the
true overall mean.
Coefficient of variation
The standard deviation divided by the mean.

4. What is Hypothesis Testing?
A  BA = B
Null Hypothesis Alternative Hypothesis

5. Example Hypotheses: Isometric Torque
• Is there any difference in the length of time that males
and females can sustain an isometric muscular
contraction?
Null Hypothesis (H0)
There is not a significant difference in the DV
between males and females
Alternative Hypothesis (HA) or experimental (HE)
There is a significant difference in the DV between
males and females.

6. • Type I errors are false positive
• results occur when a statistical significant
difference between groups is found but no
clinically important difference exists
• the null hypothesis is rejected in error

7. • Type II errors are false negative
• results a clinical important difference between
groups does exist but does not reach
statistical significance
• the null hypothesis is accepted in error
• usually occur when the sample size is small.

8. • Type 0 error: when you get the right answer,
but asked the wrong question! This is
sometimes called a type III error
• Type III error: when you correctly conclude
that the two groups are statistically different,
but are wrong about the direction of the
difference.

9. • A normal distribution is classically a bell shaped curve, that is
bilaterally symmetrical. If a variable is normally distributed,
then the mean and the median values will be approximately
equal.
• Skewness (symmetrical of the distribution)
• Kurtosis (shape of the distribution)
• outliers
can all distort a normal distribution
Testing Normality

10. Testing Normality
• A quick informal check of normality is to examine
whether the mean and the median values are close
to one another.

11. • A perfectly normal distribution has skewness and
kurtosis values equal to zero.
 Skewness values that are positive indicate a tail to the right
and skewness values that are negative indicate a tail to the
left.
 A kurtosis value above 1 indicates that the distribution
tends to be pointed and a value below 1 indicates that the
distribution tends to be flat.
 A skewness and a kurtosis values between −1 and +1
indicates normality and a value between −1 and −3 or
between +1 and +3 indicates a tendency away from
normality. Values below −3 or above +3 indicate certain
non-normality. (In Prism, 1 for skewness and 3 for kurtosis)
 Further tests of normality are to divide skewness and
kurtosis values by their standard errors. A critical value
that is outside the range of −1.96 to +1.96 indicates that a
variable is not normally distributed.

12. • In Prism, D'Agostino-Pearson normality test is
recommend. It first computes the skewness
and kurtosis to quantify how far the
distribution is from Gaussian in terms of
asymmetry and shape. It then calculates how
far each of these values differs from the value
expected with a Gaussian distribution, and
computes a single P value from the sum of
these discrepancies.
Statistical tests of normality

13. Statistical tests of normality
• The Shapiro–Wilk test has more statistical
power to determine a non-normal distribution
than the Kolmogorov–Smirnov test. It will
have a value of 1.0 for perfect normality.
• P value less than 0.05 indicates that the
distribution is significantly different from
normal.

14. Transforming skewed distributions
• When a distribution has a marked tail to the
right hand side, a logarithmic transformation
of scores is often effective.

15. One- and two-tailed tests
• One-tailed test is used to test an effect in one direction
only (i.e. mean1 > mean2) When a one-tailed test is
used, the 5% rejection region is placed only in one tail
of the distribution.
• Two-tailed test is used to decide whether one mean
value is smaller or larger than another mean value (i.e.
mean1 ≠ mean2).For a two-tailed test, 2.5% of the
rejection region is placed in the positive tail of the
distribution (i.e. mean1 > mean2) and 2.5% is placed in
the negative tail (i.e. mean1 < mean2).
• The one-tail P value is half the two-tail P value.

16. Level of Significance

17. parametric tests and non-parametric
tests
In general, parametric tests are preferable to non-
parametric tests because a larger variety of tests are
available and, as long as the sample size is not very
small, they provide approximately 5% more power
than rank tests to show a statistically significant
difference between groups.

18. Rank based non-parametric tests
Rank based non-parametric tests are used when
the data do not conform to a normal distribution.
– If the data are clearly skewed,
– if outliers have an important effect on the mean
value
– or if the sample size in one or more of the groups is
small, say between 20 and 30 cases, then a rank
based non-parametric test should probably be
used.

19. • These 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
Rank based non-parametric tests

20. • Unpaired two-sample t-test is the Mann–
Whitney U test. (These tests rely on ranking and
summing the scores in each group and may lack sufficient
power to detect a significant difference between two groups
when the sample size is very small)
• Paired t-test is the Wilcoxon signed rank test,
which is also called the Wilcoxon matched
pairs test. (In this test, the absolute differences between
paired scores are ranked and difference scores that are equal
to zero, that is indicate no difference between pairs, are
excluded from the analysis)
Rank based non-parametric tests

21. Analysis of Variance (ANOVA)
• To compare differences in the mean values of
three or more independent groups, analysis of
variance (ANOVA) is used.
The ANOVA test is called an analysis of variance and
not an analysis of means because this test is used to
assess whether the mean values of different groups
are far enough apart in terms of their spread
(variance) to be considered significantly different.

22. Concept of an ANOVA model

23. • The F value is calculated as the mean between-
group variance divided by the mean within-
group variance.
• The higher the F value, the more significant the
ANOVA test because the groups (factors) are
accounting for a higher proportion of the
variance. (Obviously, if more of the participants are closer to
their group mean than to the grand mean, then the within-
group variance will be lower than the between-group variance
and F will be large)

24. Test Matched Nonparametric
Ordinary one-way ANOVA No No
Repeated measures one-way
ANOVA
Yes No
Kruskal-Wallis test No Yes
Friedman test Yes Yes
ANOVA compares the difference among group means with
the scatter within the groups, taking into account sample size.

25. Factorial ANOVA models
• One-way ANOVA, also called one-factor ANOVA,
determines how a response is affected by one factor. For
example, you might measure a response to three different
drugs. In this example, drug treatment is the factor. Since
there are three drugs, the factor is said to have three levels.
• If you measure response to three different drugs, and two
time points, then you have two factors: drug and time. Use
two-way ANOVA.
• If you measure response to three different drugs at two
time points with subjects from two age ranges, then you
have three factors: drug, time and age. Prism does not
perform three-way ANOVA, but other programs do.

26. Post-hoc tests
• Although the ANOVA statistics show that there is
a significant difference in mean weights between
parity groups, they do not indicate which groups
are significantly different from one another.
Alternatively, post-hoc tests, which involve all
possible comparisons between groups, can be
used.
• When the F test is not significant, it is unwise to
explore whether there are any between-group
differences.

27. Post hoc test
• If you are comparing every mean with every
other mean, prism recommends the Tukey test.
• If you are comparing a control mean with the
other means, prism suggests the Dunnett's test.
• If you are comparing a bunch of independent
comparisons, prism recommends the Sidak
method, which is very similar to Bonferroni but
has a tiny bit more power.
These methods can compute confidence of interval

28. Post hoc test
If you don't care about seeing and reporting confidence
intervals, then choose the following:
• If you are comparing every column mean with every
other column mean, we recommend that you choose
the Holm-Šídák test, which is more powerful than the
Tukey method
• If you are comparing each column mean to a control
mean, Prism only offers the Holm-Šídák test.
• If you are comparing a bunch of independent
comparisons, Prism offers only the the Holm-Šídák test.

29. Analysis of covariance
• Analysis of covariance (ANCOVA) is used when
it is important to examine group differences
after adjusting the outcome variable for a
continuously distributed explanatory variable
(covariate).
• Adjusting for a covariate has the effect of
reducing the residual (error) term by reducing
the amount of noise in the model.

30. Correlation
 The correlation coefficient (r) and the nonparametric
Spearman correlation coefficient (rs). Their values
range from:
-1 (perfect inverse relationship; ax X goes up, Y goes
down) to
1 (perfect positive relationship; as X goes up so does Y).
A value of zero means no correlation at all.
 The coefficient of determination (r2) is commonly used.
For example, if r2=0.59, then 59% of the variance in X can
be explained by variation in Y.

31. Linear regression
• Linear regression finds the best line that
predicts Y from X.
• Linear regression quantifies goodness of fit
with r2, sometimes shown in uppercase as R2.
• Linear regression is usually used when X is a
variably you manipulate (time, concentration,
etc.)