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Introductory Statistics
Presentation Outline:
Types of statistics
Statistical test definitions
Simple statistical tests
What are statistics?
One way to describe statistics is as a set of
scientific techniques used for learning in the
presence of variation
Statistical measures such as P-values and
Confidence Intervals, help to quantify how
much we can learn from a sample of data
There are two types of statistics – Descriptive Statistics
Concerned with presentation, organization, and summarization
of data
Give a lot of VERY important information about data prior to
performing Inferential statistics (%’s, means, confidence
intervals)
Inferential Statistics
Used to make inferences about the characteristics of the
population from the characteristics of a random sample drawn
from the population
Hypothesis testing: using data samples to establish the
credibility of a theory about the population
P values are calculated from the different inferential statistical
tests to confirm the study hypothesis
Inferential Statistics Inferential statistics are then divided into two more
categories:
Parametric - assumes a normal distribution based on
population means and standard deviations
• Interval/Integer (pain level, temperature)
• Ratio (weight-Body Mass Index)
Nonparametric - make no assumptions about the nature
of the distribution underlying the data; these statistics are
not distribution free, we do not know what the distribution
looks like
• Nominal (gender, ethnicity)
• Ordinal (tumor position)
Inferential Statistics cont’d.
The type of inferential statistical test performed is driven by the
type of data being analyzed
Types of Data
Categorical / Nominal data consists of named categories with
no implied order among the categories.
Ordinal / Rank data consists of ordered categories, where the
differences between categories cannot be considered to be
equal.
Continuous data may take any value within a defined range and
assumes equal distances between values.
Choosing an appropriate test
Assumptions
inferential tests have certain assumptions that
you should be familiar with before you use a
test
violating the assumptions misleading results
Considerations for choosing a test
• Variables
• Distribution
• Parametric and non-parametric
The type of research design and type of data will ultimately drive the appropriate statistical test used (if all assumptions of the statistical test are met).
Type of Statistical Test
Type of Data
Continuous Ordinal Categorical
Type of Design Parametric Non-parametric
Compares 2 Independent groups Independent t-test Wilcoxon-Mann-
Whitney test Chi-square test (r x 2)
Compares 3 Independent groups One-way ANOVA Kruskal-Wallis test Chi-square test (r x k)
Compares pre and post in the same
sample size Paired t-test
Wilcoxon signed rank
test McNemar Change test
Compares multiple measures in the
same sample size
Repeated Measures
ANOVA Friedman test Cochran Q test
Correlation between two variables Pearson Correlation Spearman Correlation Kappa coefficient
To model which variables predict an
outcome Multiple Regression
Multinomial Logistic
Regression Logistic Regression
Chi-Square Test (Pearson’s, goodness-of-fit)
Underlying concept: do the observed frequencies
differ from the expected frequencies?
If H0 (NULL) is true: expected = observed
If HA (ALTERNATIVE) is true: expected ≠
observed
Design is represented by contingency tables (often
stated in % age, but count is level of analysis)
Contingency tables = frequency tables = cross-
tabulation tables
Generic Contingency Table
For a 2 x 2 contingency table the Chi-square statistic is calculated
by the formula:
Just like with the t-test, the computation will result in a test
statistic and the associated p-value that will allow discussion of
the group differences.
A B
C D
A + C B + D
A + B
C + D
Fischer’s Exact Test
Similar to the chi square test, but is preferred with:
Smaller sample sizes
Severely unequal cell distribution
Cells with an expected frequency of < 5 OR 10
T-test
Independent t-test / Student’s t-test – compares
continuous data (means) between 2 independent
groups (most robust of all statistical tests)
Paired t-test – compares continuous data (means)
between 2 dependent / matched / paired groups
ANOVA 3 or more groups
Multiple repeated measures
Within and between subject designs (also MIXED
design)
Study Design (one-way ANOVA analysis)
• Group A: full dose of drug ‘wonderful’
• Group B: half-dose of drug ‘wonderful’
• Group C: placebo
• t-test = 4 tests, ANOVA = 1 test to tell if a difference
exists
Two-way ANOVA
Two independent variables (IV) (Example: 2x2, DV – BMI)
Main effect of Diet: Yes (A+B) vs. No (C+D)
Simple Main Effect of Diet Yes: A vs. B
Interaction: A by B by C by D
(weight loss) DRUG
Diet Plan
YES NO
YES 20 (A) 26 (B)
NO 23 (C) 29 (D)
Repeated Measure ANOVA
Within subject design only or within –between
design
Every subject is exposed to each level of an
IV
Most common example: time
All other types of repeated measure are less
common and not appropriate for many
questions
Tip: avoid order effects
Measures of Association Pearson’s Correlation
A measure of the strength of association between 2
variables
Ranges from -1 to +1,
Correlation coefficient is r
-/+ signs indicate the direction of the relationship
≠ causation
0.10 (small), 0.30 (medium), 0.50 (large)
Spurious/illusory correlations
Statistical significance does not mean the results are
clinically significant
Confidence Intervals (CIs)
A range of values within which a researcher can
say with a certain degree of confidence that a
population parameter will fall.
Originally designed to analyze a sample of samples, but
is now used on one sample
Useful when the mean is uncertain do to conflicting
results
Meta-analysis
Used to test non-inferiority and superiority
Provides confidence without specificity
CIs and Hypothesis Testing
Factors That Create Misleading Results
Restricted range tends to reduce r
Nonlinear relationships – cannot use
Pearson’s correlation
X and/ or Y have skewed distribution –
underestimate r
Outliers – over- or underestimate r
Extreme groups – overestimate r
Non-Normal distribution Spearman rank correlation
Uses the same ranking principle as the Mann-Whitney
Same characteristics as Pearson’s r
• Ranges from -1 to +1,
• Correlation coefficient is rs
• -/+ signs indicate the direction of the relationship
• ≠ causation
• Spurious correlation
• Statistical significance does not mean clinically
significant
No correlation for dichotomous data (chi-square)
Non-parametric group comparisons Mann-Whitney U Test
Alternative to the independent t-test when normal
distribution is severely violated
Converts raw scores to ranks
Compares ranks between groups to determine if there is
a difference
Wilcoxon Signed-Ranks Test
Alternative to the dependent t-test when normal
distribution is severely violated
Compares rankings of difference scores
References UCLA
http://www.linguistics.ucla.edu/faciliti/facilities/statistics/power.htm
The Florida State University
http://stat.fsu.edu/undergrad/statinf2.php
About.com Sociology
http://sociology.about.com/od/Statistics/a/Descriptive-inferential-
statistics.htm
University of South Carolina
http://www.usca.edu/polisci/apls301/Text/Chapter%2012.%20Significance
%20and%20Measures%20of%20Association.htm
Boston University School of Public Health
http://sphweb.bumc.bu.edu/otlt/MPH-
Modules/BS/BS704_Nonparametric/BS704_Nonparametric2.html
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