Lectures of basics and data and selecting a test. Compliments to multiple online resources used in the creation of this PowerPoint.

1. Analysis 101
Brian Wells

2. Today’s Objectives
• Not to teach you the mathematics involved
• Not to make you an expert statistician
• Not to make you an expert in picking tests and
designing studies
• Is to highlight different analytic and statistical
methods in research
• Is to help facilitate communication between
investigators and biostatisticians by establishing a
common vocabulary

3. Data Types
• Numerical data (quantitative)
• Measurements or counts
• Weight, blood pressure, number of medications
• Categorical data (qualitative)
• Patients sorted into categories
• Diabetic/non-diabetic
• Adherent/non-adherent
• Smoking/non-smoking

4. Categorical Data
• Nominal
• No explicit ordering to categories
• Blood types – A/B/AB/O
• Race/Ethnicity
• Called binary or dichotomous if 2 categories
• Gender – M/F
• Ordinal
• Defined ordering
• Cancer stage I, II, III, IV
• Non-smoker/smoker/ex-smoker
• NYHA Class

5. Numerical Data
• Can be further subdivided into discrete and
continuous
• Discrete variables
• Have a limited number of possible values (finite or
countably infinite)
• Gaps between possible values (whole integers)
• Ex: Number of CHF episodes, number of medications
• Continuous variables
• No gaps between possible values
• Ex: Duration of seizure, body mass index, height

6. Determining Data Types
• Ordinal (Categorical) v. Discrete (Numerical)
• Ordinal
• Cancer Stage I, II, III, IV
• Cancer Stage II is not 2*Stage I
• Discrete
• Number of children: 0, 1, 2…
• 4 children = 2 times 2 children

7. So Why Spend Time On This?
• The data types help determine which analysis to
use
• It helps determine how best to summarize the
display data
• Categorical – percent's, fractions, numbers in
categories
• Numerical – mean, median, mode, standard
deviation, variance, quartile ranges

8. Data Summaries
• Be careful of overreliance on numbers – Keep the
big picture in mind (more on this next time)
• Both means = 2, SD = 1.9, n = 1000

9. Statistical Inference
• Estimation of quantity of interest
• Estimate itself
• Quantify how good an estimate it is
• Ex: If you took more and more samples, how much
would the estimate vary?
• Hypothesis testing

10. Statistical Inference Example
• Proportion of people in a population who have diabetes.
N = 800
• Sample 1: 200/800 = 0.25
• We conclude that the estimated % of people with
diabetes is 25%
• But how variable is our estimate?
• We need to know the sampling distribution!
• Option 1: Take lots and lots of samples
• Sample 2: 215/800 = 26.8%
• Sample 3: 194/800 = 24.25%
• Not practical!

11. Statistical Inference Example
• Statistical theory
• Sample distributions for means and proportions are
normally “bell-shaped”
• From a single sample, we calculate the standard error
(variability) of our estimated mean or proportion
• Standard error measures the variability of the sample
statistic. Small SE means more precise estimate.
• SE ≠ Standard Deviation
• SD = variability of the sample data
• SE = variability of the statistic

12. Distributions
• Sample means follow a t-distributions on if
• Underlying data is approximately normal OR
• N is large
• A sample mean from a sample of size n will have a t
distributions with n-1 degrees of freedom (tn-1)

13. Confidence Intervals
• Assume we use our t15 distribution with n = 16, mean SBP
= 123.4 mm Hg, and SD = 14.0 mm Hg
• SE of mean = SD / √n = 3.5
• 95% CI for sample mean is then
• Mean + 2.131 (for t15 distribution) * SE
• = 123.4 ± 2.131 * SE
• = (115.9, 130.8) mm Hg
• And as N gets larger, t statistic gets smaller (t99 = 1.984),
which with the same numbers as above but with N = 100,
CI narrows to (120.6, 126.2)
• Note: It’s never incorrect to use a t-distribution as long as
the underlying population is normal or N is large

14. Hypothesis Testing
• Confidence intervals told us the best estimate and the
variability of the best estimate
• Hypothesis testing tells us if there really is a difference
between an observed value and another value
• From our earlier example: N = 800, we estimated that
25% of people had diabetes
• Let’s say a study 10 years prior estimated that 12% of
people had diabetes
• Has the percent of people with diabetes really changed?

15. Hypothesis Testing
• Support the true percent of people with diabetes is 12%
• Called the null hypothesis or H0
• How likely is it that we would observe a result as or more
extreme than 25% given the true percent is 12%?
• This is the p-value, computed using normal distributions for
sample proportions and t-distribution for sample means
• If the probability is small, consult the supposition may not be
right
• Reject the null hypothesis in favor of the alternate
hypothesis Ha
• If the probability is not small, conclude that there is
insufficient evidence to reject the null hypothesis
• This is NOT the same as accepting the null or showing the
null hypothesis is true

16. Hypothesis Testing
• H0: True proportion is 12%
• Ha: True proportion is not 12%
• If P < 0.05, we would conclude it is not likely to observe
our data is the true proportion was 12%
• We conclude that this is sufficient evidence that the
proportion with diabetes is not 12%
• Test can be one-sided or two-sided
• One-sided ONLY ok if previous research suggests that the
proportion is larger

17. Misinterpreting the p-value
• A p-value of 0.32 (or > 0.05) DOES NOT mean:
• We accept the null
• There is a 32% chance the null is true
• It only lets us reject the null in favor of the alternative or
fail to reject the null
• If you fail to reject, it DOES NOT mean the alternative isn’t
true. It may mean your N is too small or the study is
underpowered.

18. Decision-Making

19. Other Statistics
• Some statistics are distribution-free
• Recall that t-tests/distributions depend on normality or
large N’s
• What is we don’t have one or both of these, ex: skewed
data, N is small
• We can use nonparametric methods that look at ranks,
not means
• The median is a nonparametric estimate

20. Nonparametric Methods
• Don’t require a particular distribution
• Well-suited to hypothesis testing
• Not as useful for point estimates or Cis
• Especially useful is data is ranks or scores – Apgar scores,
Vision (20/20, 20/40)
• Do inferences on medial values
• Hypothesis Test is Sign Test
• Assumes hypothesized value of median is correct,
except to observe about half the sample above and
half below
• Computes probability for proportion above median

21. Parametric v. Nonparametric
• Nonparametric are always ok to use
• Nonparametric are more conservative than parametric
• In fact, 95% CI for medians are sometimes twice as
wide as those for the mean
• If your N is fairly large, or if you know your data is normal,
parametric is always best

22. How To Select A Test
• Start by asking, “Am I testing for a difference or a
relationship in my data?”

23. Difference Testing
• Am I testing one sample or more than one sample?
• One sample – Is my data parametric?
• Yes – One sample t-test
• No – Wilcoxon Signed Rank Test

24. Difference Testing
• More than one sample – Is my data nominal, or
ordinal/interval/ratio?
• Nominal – Chi-Squared test
• Ordinal/interval/ratio – How many dependent
variables are there?
• Two or more – Multivariate Analysis of Variance
(MANOVA)

25. Difference Testing
• One – Are the measures repeated, independent, or
mixed?
• Mixed – Mixed Model ANOVA
• Independent
• How many conditions are there?
• Two conditions
• Parametric data – Independent samples t-test
• Non-parametric data – Mann-Whitney U test
• More than two
• Parametric – Between Participants (One-Way)
ANOVA
• Non-parametric – Kruskal-Wallis
• Repeated

26. Difference Testing
• One – Are the measures repeated, independent, or
mixed?
• Repeated
• Two Conditions
• Parametric – Paired Samples t-test
• Non-parametric – Wilcoxon Matched Pairs
• More than two conditions
• Parametric – Within Participants ANOVA
• Non-parametric – Friedman’s ANOVA

27. Relationship Testing
• Single Independent variable
• Parametric – Pearson’s Correlation
• Non-parametric – Spearman’s Correlation
• Multiple Independent variables
• Parametric – Logistic Regression
• Non-parametric – Multiple Regression
• Multiple Factors Correlation Matrix
• Factor analysis

28. Model Information
• The specific of each model (how they differ, how they’re
calculated, etc) are not important for our purposes
• What is important is to be able to select the correct test
• Selecting the wrong test WILL lead to wrong conclusions
(failing to reject the null, inappropriately rejecting the
null)

29. Going Further
• There are many, many more tests we did not cover
• Durbin-Watson
• Kolmogorov-Smirnov
• Anderson-Darling
• Cox Proportional Hazards
• Kaplan-Meier Survival Analysis
• And so on…
• However, the tests presented will cover the majority of
basic studies done