Title: Introduction to Clinical Biostatistics for Medical Students
1Introduction to Clinical Biostatistics for
Medical Students
- Atif Zafar, MD
- Department of Medicine
2Overview of Presentation
- Introductory Concepts (Review)
- Hypothesis Testing
- Linear Regression and Correlation
- Analysis of Variance (ANOVA)
- Nonparametric Statistics
- Survival Analysis
3Introductory Concepts
4Introductory Concepts
- Types of Data
- Presenting Data
- Descriptive Measures
- Probability and Distributions
- Estimation Techniques
5Types of Data
- Data are usually Discrete or Continuous
- Discrete Variables take on a finite set of values
that can be counted - Race, Gender, Year in School etc.
- Continuous Variables take on an infinite set of
values - Age, Height/Weight, Blood Pressure
6Types of Data
- A Special type of Discrete Variable is the Binary
Variable which takes on exactly 2 possible values - Gender (M/F)
- Pregnant? (Y/N)
- Hypertensive? (Y/N)
7Types of Data
- Sometimes, discrete variables have a natural
ordering to them - For example, names of consecutive days in a week
(M, Tu, Wed, Thurs, Fri, Sat, Sun) - Other types of discrete variables do not have a
natural order and are called Nominal Variables - Race (African American, Caucasian, Asian,
Hispanic etc.)
8Types of Data
- If in an experiment you measure a single
variable, it is called a Univariate experiment - If you measure 2 variables, it is called a
Bivariate experiment - And if you measure multiple variables, it is
called a Multivariate experiment
9Types of Data
- A Random variable is one whose value is
determined by chance or random event - Typically, a variable X is random if it is the
outcome of an experiment where results can occur
by chance or are not completely predictable
10Types of Data
- Nonparametric Variables
- Many times in clinical studies, we seek opinion
data (I.e. patient satisfaction scores, relative
value scales etc.) - The data can be ranked but has no absolute scale
that is comparable - This type of data is called nonparametric data
11Presenting Data
- There are many ways to present data
- Frequency Tables
- Pie Charts
- Bar Graphs (Histograms)
- Line Graphs
- Scatter Plots (Scattergrams)
- Stem and Leaf Displays
- Box Plots
12Presenting Data
- Scatter Plots (Plot of a Bivariate experiment)
13Presenting Data
- Stem and Leaf Displays
- Presents a histogram like picture of the data,
while retaining the original data values - Dataset 8520 9274 8142 11298 10624 7987
11172 12899 10737 9198 13625 9462 11847
10178 12240 11690 10069 11240 12745 12995
14Presenting Data
- Boxplots
- Complex visual data structures that combine
various measures - Maximum and Minimum Data Points
- 1st and 3rd Quartile Points
- Sort the data points from lowest to highest
- Divide the number of data points into 2 halves
- Take the Median value of each half and those are
the 1st and 3rd quartiles (Q1,
Q3) - Computer the Inter Quartile Range (IQR)
- IQR Q3-Q1
- Compute 1.5 x IQR. Compute Q31.5IQR and
Q1-1.5IQR - Data points lying outside this range are called
Outliers
15Presenting Data
16Descriptive Measures
- Now that we have displayed our data, we want to
be able to characterize it quantitatively - Measures of Central Tendency
- Mean, Median, Mode
- Measures of Variability
- Range, Variance, Standard Deviation
- Measures of Relative Standing
- Z-Scores, Percentiles, Quartiles
17Measures of Central Tendency
- Mean
- Arithmetic Average of a sample of data
- Median
- If you order the data from smallest to highest,
the median is the middle value, assuming an odd
number of data elements - If you have an even number of elements, it is the
average of the 2 middle numbers. - Mode
- The most common value in a set of values
18Measures of Variability
- Once we have located the center of a set of data
points, we want to know how dispersed they are
19Measures of Variability
- Range
- This is the difference between the highest and
lowest value - Variance
- Defined to be the average of the square of the
deviations of the individual data points about
their mean - Standard Deviation
- This is defined as the square root of the
variance
20Measures of Relative Standing
- Sometimes we want to know the position of a
particular observation relative to others in a
data set - Ex How you performed with respect to your
classmates on an exam - The Z-Score measures this as follows
21Measures of Relative Standing
- Percentiles and Quartiles also indicate relative
standing but in terms of the categories of scores
from lowest to highest - Given a set of n measurements x1, , Xn the pth
percentile is defined to be the value of x that
exceeds p of the measurements and is less than
(100-p) of the values. - Ex Scores of 20, 30, 50, 60, 67, 67, 70, 80,
90, 95 - The score 50 is in the 30th percentile, meaning
that 30 of the scores were lower than yours and
70 were higher than yours. - Quartiles similarly reflect in which quarter of
the set of values a particular observation lies - Ex Scores of 20, 30, 50, 60, 67, 67, 70, 80, 90,
95 - 1st Quartiles 50, 3rd Quartile 80
22Probability
- Suppose you do an experiment with a finite number
of possible outcomes (ex coin toss) - The Probability of an event E (H/T) is the chance
() that the event will turn out in a given way
in the next repetition of the experiment - Probabilities values are always between 0 and 1
- The notation for probabilities is as follows
- Given our coin toss experiment,
- P(H) Probability that a Head will be tossed in
the next round - P(T) Probability that a Tail will be tossed in
the next round - One can estimate probabilities by repeating the
event many times and observing the outcomes
23Probabilities Some Simple Rules
- Arithmetically, one can combine probabilities of
simple and sequential events - Given a complex event composed of N simple
events, the probability of the complex event is
equal to the sum of the probabilities of each of
the simple events - Ex Coin toss 1 and Coin toss 2
- Event First Coin Second Coin P(Ei)
- E1 Heads Heads ¼
- E2 Heads Tails ¼
- E3 Tails Heads ¼
- E4 Tails Tails ¼
- Let A E2, E3. Then P(A) P(E2)P(E3) ½
24Probability Distributions
- Given a random variable X (either discrete or
continuous), the Probability Distribution gives a
table or formula or graph of the probabilities of
each potential value of X - For a Probability Distribution P(x) the following
must hold - 0 lt P(x) lt 1
- Sum (all P(x) over all x) 1
25Probability Distributions
- There are many kinds of probability
distributions - Binomial Distribution
- Applies to binary variable experiments where only
2 outcomes are possible - Poisson Distribution
- Applies to variables that represent the number of
occurrences of a specified event in a given unit
of time or space - Hypergeometric Distribution
- Applies to experiments where the numbers of
elements in the population is small in comparison
to the sample size and thus the success of a
trial depends on the outcomes of preceding trials
26Probability Distributions
- Normal Distribution (N)
- Applies to continuous random variables
- Standard Normal Distribution (Z)
- A Normal Distribution with
- Mean of 0
- Standard Deviation of 1
27Estimation Techniques
- So now that we know that certain experiments
can have results distributed in certain ways, how
can we predict the result of this experiment? - This process is called Statistical Inference,
where we can estimate the quality of a larger
population by analyzing a small sample
28Estimation Techniques
- Populations and Samples
- A Population is the larger set of objects we wish
to study - Ex The number of democrats in the country
- A Sample is a set of representative objects we
choose in order to estimate the characteristics
of the larger set of objects - Ex Take 100 people from each state and determine
whether they are democrats
29Estimation Techniques
- Parameters and Statistics
- A Parameter is the quality of the population we
are trying to estimate - In order to estimate the parameter we measure the
quality in a sample. This sample quality is
called its statistic
30Estimation Techniques
- Many types of samples can be taken
- Completely Random Sample
- Stratified Random Sample
- Divide the population into strata (groups)
- Take a sample from each group
- Ex Party loyalties of teenagers, adults and
elderly - Cluster Sample
- Take a simple random sample of clusters from the
available clusters in a population - Ex Urban vs. Rural sampling
31Hypothesis Testing
- Large Sample Estimation Techniques
32Introduction to Estimating Techniques
- Before we begin, lets review some common terms
- Point Estimate When we do an experiment and
generate a result, the result at one point in
time for one run of the experiment is called a
point estimate (mean, etc.). Since each
experiment has some error, there is a margin of
error for every point estimate - Interval Estimate Now if we repeat the
experiment many times over we will get sense of
how far off we are from running a perfect
experiment. This sense of confidence in our
experimental ability is called an interval
estimate or a confidence interval.
33Confidence Intervals
- Typically, the confidence interval is defined as
follows - CI Mean /- 1.96 x Variance / sqrt(N)
- It tells us that if we repeat the experiment
many times over, 95 of the time our values for
the Mean will lie in the limits specified here
34Significance Value (a)
- Statisticians arbitrarily choose a value of 5 to
represent events that can occur by chance alone - So if an event occurs more than 5 of the time,
it is considered statistically significant - The 5 value is called a significance value, or a
35P-Values
- A P-value is a useful way to represent the
probability of a certain event and is seen
extensively in the medical literature - Definition
- The P-Value is simply the probability that an
event occurs by chance alone - Given our significance level of 5 for chance, we
want P-values to be less than 5 or .05 to be
considered statistically significant
36Comparing Means
- Many times we wish to compare the means of two
subsets of a population - Ex MCAT scores for Biology vs. Chemistry majors
- To do this we would sample MCAT scores from
random samples of biology and chemistry majors
across the country - We would compute the mean of all these samples
- We would compare the means to determine if they
are significantly different. - This kind of analysis is exactly what is done by
Hypothesis Testing (we hypothesize there is no
difference and then refute this hypothesis)
37Hypothesis Testing
- A statistical test of hypothesis consists of 4
parts - A NULL Hypothesis, termed Ho
- An Alternate Hypothesis, termed Ha
- A test statistic
- A rejection region
- The NULL hypothesis is what we want to refute
- The Alternate hypothesis is what we want to
support - The test statistic is what we will use to compare
the NULL and the Alternate Hypotheses - The Rejection Region is the value of the test
statistic for which Ho will be rejected
38Hypothesis Testing
- So what does this all mean IN LAYMANS TERMS?
- Basically we are asking the question that given a
test statistic we specify, what is the
probability that the hypothesis in question (Ha)
is due to chance alone? - We convert the test statistic into a probability
value by looking it up in a table that specifies
the respective probabilities associates with that
particular statistic value
39Constructing a Hypothesis
- Consider the following question
- We wish to show that the hourly wages of
construction workers in California is larger than
the national average of 14 - The hypothesis will be written down as
- Ha ? ltgt 14
- Ho ? 14
- Test statistic Z-value X Uo /
(Var/sqrt(N)) - Rejection region 0.05 (a value)
40Testing a Hypothesis
- The average weekly earnings for men in managerial
and professional positions is 725. Do women in
the same position have average weekly earnings
that are less than those for men? - A random sample of N40 women in managerial
positions showed X670 and Var 102. Test the
appropriate hypothesis using a 0.01 - Solution Ho U 725 Ha U lt 725
- Z X U / (Var/sqrt(N))
- Z 670 725 / (102 / sqrt(40)) -3.41
- Since -3.41 lt 0.01 we conclude that Ho is false
and the average weekly salary for women is
significantly less than for men and the
probability that we have made an incorrect
decision is 0.01
41Confidence in our Test Result
- So what is our confidence in our result?
- Well, we can have 2 types of errors
- Type I error Rejecting Ho when Ho is true a
- Type II error Accepting Ho when Ho is false b
- To compute a confidence value, we calculate the
Power of the Test which is the probability of
correctly rejecting the NULL hypothesis - Power (1-b)
42Types of Tests
- Given the kinds of data we have and the types of
information we seek there are different types of
tests available to us - Students T-Test
- Used to compare MEANS of two populations
- Works for small samples (Nlt30)
- Chi-Square Test
- Used to estimate a populations VARIANCE
- F-Test
- Used to compare the VARIANCES of 2 populations
43Types of Tests
- We can do these tests in different ways
- We can have one-tailed and two-tailed tests
- A One-tailed test occurs when our hypothesis mean
is on one side (either less or greater) than the
null hypothesis mean - A Two-tailed test occurs when we can say that the
hypothesis mean can be on either side of the null
value - We can also do Paired Tests, where we do 2 tests
in a specific sequential order
44T-tests Small Sample Testing
- Up to now we have assumed the sample size to be
large (Ngt30) in order to achieve good power. But
what happens when the sample size is small
(Nlt30). - Well, in this case the shape of the normal
distribution looks somewhat different it is
shorter and wider and is called the
T-Distribution - Every T-distribution has an associated Degree of
Freedom (df) which is equal to N-1 - A T-Table is consulted to get the appropriate
values of the T-statistic when doing a T-test.
You need the df and the significance level to
look up the T-values.
45Chi-Square Distribution
- Remember that the T-test compares population
Means. What if we want to estimate a population
variance? - In this case, we would use a Chi-Square
distribution and our test statistic will be a
chi-square value - X2 (n-1)s2 / oo2
- where n sample size
- s sample variance
- oo Population Variance that we are trying to
estimate - A variant of the Chi-Square Distribution is
called the Mantel-Haenszel Test - It is a test of association between 2 ordinal
variables (frequency data)
46F-Distribution
- What if we want to compare the population
variances of two different populations? - In this case we use an F-Distribution and an
F-statistic -
- F s12/s22, where s1 and s2 are variances of
Samples 1 and 2 - Typically we will have 2 degrees of freedom (v1
and v2) with F-tests
47Linear Regression and Correlation
48Linear Regression and Correlation
- In many situations in clinical studies we wish to
attempt to answer the question How is the
random variable X related to the random variable
Y? - Ex How is smoking related to lung cancer?
- Ex How is age related to development of
Alzheimers Disease? - Ex How is hypertriglyceridemia related to
metabolic syndrome? - Such questions are answered statistically using
the concepts of Regression Analysis which looks
for relationships among different variables
(either negatively or positively) and
Correlations, the strengths of the relationships - Relationships may have many forms
- Related linearly
- Related curvilinearly
- Related colinearly
- Associations but not Correlations
49Linear Regression
- The Linear Regression model postulates that two
random variables X and Y are related by a
straight line as follows - Y a bX e
- Where
-
- Y is the dependent variable
- X is the independent variable
- a is the intercept
- b is the slope
- e is the residual value
50Linear Regression
- Residual Value (e)
- Given that the regression analysis procedure is
itself a statistical approach, it is expected to
have some degree of error associated with it - Thus we add a value called the residual value (e)
to any regression equation to account for random
errors in the process - Scatterplots
- In order to perform regression analysis visually,
it helps to graph the points on a scatterplot - A visual relationship can often be observed when
looking at these plots
51Method of Least Squares
- So, assuming that 2 variables are linearly
related, how do we best fit a line through a
series of points on a scatterplot the
regression line. - One way is to use a goodness of fit estimator
called the Sum of Squares for Error (S) which we
want to minimize -
f(xi)
yi
52Inferences Concerning Slopes
- The initial question once we have a regression
line is whether the data present sufficient
evidence to indicate that Y increases or
decreases linearly as X increases over the
observed region? - So we use the variability of the points about the
line to estimate this - Variance s2 S / n 2
- S Sum of squares for error
- n Sample size
53Inferences Concerning Slopes
- Given that we can use S for estimating the
population variance, we can formulate our
hypothesis using a T-test to compare means as
follows - Null Hypothesis Ho b bo
- Alternate Hypothesis Ha b lt bo or b gt bo
- Test Statistic t-value b bo / (s /
sqrt(Sxx)) - b regression line slope
- bo slope to test with
- s variance
- Sxx Standard Error for Xis Sum over all i
(Xi Xmean)2
54Inferences Concerning Slopes
- So how do we do the T-test and reach a conclusion
or calculate a P-value? - Well, the T-table has several features
- Df Degrees of Freedom n 1
- T-values listed for various significance levels
- The procedure for using a T-Table is as follows
- Compute the T-value using the statistic in your
test - Lookup the appropriate T-value in the table given
your degree of freedom (n 1) - Then look up the column to whichever significance
level it belongs to and the P will be less than
that significance level
55Linear Regression
- So, graphically what does it look like?
56Other Regressions
- Given the types of data you have, there are other
methods for fitting the data to a geometric
shape - For example, there is Curvilinear Regression
- Cubic Spline Interpolation
- Quadratic Interpolation
- Higher Order Interpolation
- Logarithmic Regression
- This is useful when you have categorical data
(non-numeric) - For example, when you have a binomial random
variable such as HTN (y/n), Gender(M/F) or Race
57Correlation
- As opposed to finding the best fit line through
a set of data points, Correlation seeks to
understand the strength of the relationship. - R 0.17 R 0.85 R -0.94
58Correlation
- We compute the Pearson Product Moment Coefficient
of Correlation (R) as follows - R Sxy / sqrt (Sxx X Syy)
- where
- Sxy Sum over all i (Xi Yi)2
- Sxx Sum over all i (Xi Xmean)2
- Syy Sum over all i (Yi Ymean)2
- 0 lt R lt 1, the larger the R the stronger the
correlation
59Multiple Linear Regression
- So far we discussed how one variable is related
to another in a study. - But in real life, a study typically has many
variables that it is trying to compare as they
related to an outcome - Ex CAD as f(HTN, DM, Smoking, Hyperchol.,
Obesity, Age) - In order to do this type of analysis, we can
extend the general notion of linear regression to
multiple variables. - We have an intercept as usual but partial slopes
(or partial regression coefficients), each one
representing a different variable
60Multiple Linear Regression
- The General Linear Model (GLM) is then stated as
follows - Y b0 b1x1 b2x2 b3x3 .
bnxn e - With the following assumptions
- 1. Y is the response variable you wish to
predict - 2. b0, b1 . bn are unknown constants
- 3. x1, x2 . xn are independent predictor
variables that are measured without error - 4. e is a random error that for any set of
predictors is normally distributed - 5. The random errors associated with any pair of
Y values are independent
61Multiple Linear Regression
- Note that you can use qualitative (categorical)
and quantitative variables in a GLM. - Categorical Variables look like
- X1 1, if Group A, 0 if not Group A
- Typically computing p-values and regression
equations in a GLM is hard to do by hand so most
people will do it using computer software - SAS has a procedure called Proc GLM
- SPSS/PC
- StatSoft
- HyperStat
62Multiple Linear Regression
- Problems that can occur when using GLM
- Multicolinearity
- This happens when 2 of the independent variables
xi, xj are themselves related and occurrence in a
model overestimates the true effect size - Also known as Covariants or Confounding Factors
- Interaction Terms
- When 2 variables in a model are co-related then
we must add an interaction term to the model - For example, suppose you want to study the salary
of a professor with respect to of years of
service. Well, this may differ slightly whether
you are a male or female. - Thus, the salary slope for males may be slightly
higher than the salary slope for females despite
the same number of years of service. - This type of relationship is called an
Interaction (between gender and years of service
because the slope varies depending on whether a
male or female is selected) and we must add a
term of the type - Y b0 b1x1 b2x2 b3x1x2
63Logistic Regression
- What happens when you have data in the form of
proportions (or frequency data) of categorical
variables? - The tool for analysis of this type of data is
called a Logistic Regression - It is based on the Chi-Square Distribution and
the model is described as follows - lnp/(1-p) a BX e or p/(1-p) expa
expB X exp e - where
- ln is the natural logarithm, logexp, where
exp2.71828 - p is the probability that the event Y occurs,
p(Y1) - p/(1-p) is the "odds ratio"
- lnp/(1-p) is the log odds ratio, or "logit"
- all other components of the model are the same.
64The ANalysis Of VAriance
65ANOVA
- Suppose you want to compare the mean
reimbursement rates from 5 different health plans - You could do t-tests among all combinations of
the 5 plans, or 10 t-tests - Suppose all the means are equal. When this
procedure is repeated 10 times, the probability
of incorrectly concluding that at least one pair
of means differ is quite high and you reach an
erroneous decision - Thus we want one test which could compare means
for all 5 groups at the same time - This is exactly what ANOVA provides
66ANOVA
- ANOVA is a powerful procedure which allows you to
do 2 things - Compare the variance between the means of 2 or
more groups - Compare the variance in data values within each
group
67ANOVA
- ANOVA procedures can be done with different study
designs - Completely Randomized Design
- Random samples are independently selected from
each of k populations. - Assumes that the data is homogeneously
distributed with a fixed variation - Randomized Block Design
- Assumes that subsets of the population have
different variances - Within each subset, however, the variability is
the same - Each subset is called a block.
- Random samples are then taken from each block
68ANOVA for Completely Randomized Designs
- Suppose we want to compare k population means
u1..uk based on random independent samples of
n1..nk observations selected from populations
1..k respectively - Ex Suppose we have 10 observations of
reimbursement figures from each of 5 health plans
then we will have 50 total values - Then let
- Xij represent the jth measurement in the ith
group - We define an entity called the Total Sum of
Squares (SS) as follows - k ni
- Total SS Sxx ? ? (xij x)2
- i1 j1
69ANOVA for Completely Randomized Designs
- It can be shown that the sum of squares of
deviations of all values about the overall mean
the Total SS - (of all 50 values) can be
partitioned into 2 components - SST Sum of Squares for Treatments
- SSE Sum of Squares for Error (measures
variation within samples) - We have
- Total SS SST SSE
70ANOVA for Completely Randomized Designs
- Now, we can also compute SSE readily and it is
- n1 n2 nk
- SSE ? (x1j x1)2 ? (x2j x2)2 ? (xkj
xk)2 - j1 j1 j1
- Knowing SSE and SS, we can compute SST
- We then compute the Mean Squares of these as
follows - MST SST / k-1
- MSE SSE / n-k
- The final step is to compute an F-statistic as
follows - F MST / MSE
71ANOVA for Completely Randomized Designs
- Now, F-tests have 2 degrees of freedom v1 and v2
- In the case of ANOVA,
- v1 k 1
- v2 n k
- We can then our usual hypothesis testing using
this F-statistic as our test - Ho u1 u2 u3 uk
- Ha One of more pairs of population means differ
- F-Statistic MST/MSE with df v1(k-1), v2(n-k)
- Rejection Region Reject Ho if F gt Fa (found
from the table using v1, v2 and a)
72ANOVA for Randomized Block Designs
- The computational steps are very similar to those
of a completely randomized design except that we
add a third term, the sum of squares for BLOCKS
(with b blocks) - Total SS SST SSE SSB
- We then perform 2 different hypothesis tests
- (1) For comparing Treatment Means
- F MST/MSE, v1k-1, v2n-b-k1
- (2) For comparing BLOCK Means
- F MSB/MSE, v1b-1, v2n-b-k1
73Nonparametric Statistics
74Nonparametric Statistics
- What do we do when we have oppinion data?
- For example, suppose a judge is employed to
evaluate and rank the sales abilities of 4
salesmen, the edibility of 5 brands of Corn
Flakes or the relative appeal of 5 brands or
automobiles - Clearly it is impossible to give an exact measure
of sales competence, the palatability of food or
design appeal - But, it is possible to rank the salespeople, food
or design choices based on our own oppinions. - Many, Many types of studies in medicine use this
kind of data gathering (patient satisfaction is
one example)
75Nonparametric Statistics
- There are many tests available for studying this
kind of data - The Sign Test
- The Mann-Whitney U Test
- The Wilcoxon Signed-Rank Test for a Paired
Experiment - The Kruskal-Wallis H Test for Completely
Randomized Designs - The Friedman Fr Test for Randomized Block Designs
- Spearmans Rank Correlation Test
76The Sign Test
- Compares 2 populations with respect to how they
differ in the responses to qualitative questions - Compute the number of responses that were the
same - Then compute the number of responses that
differed - Finally compute X, the number of times responses
from population A was greater than responses from
population B - This gives you the number of times (A-B) is
positive (i.e. has a positive sign hence the
name) - This is your test statistic
- You then use a Binomial Probability Distribution
to do a hypothesis test
77Mann-Whitney U Test
- Analogous to the T-test for nonparametric data
- Suppose you have 2 populations from which 2
samples n1 and n2 are obtained - You should rank all samples (n1n2) into
ascending order assigning rank values 1, 2, 3 to
all observations - Tied observations are handled by averaging the
ranks assigned to both of the tied observations - Then calculate the sum of the ranks T1 and T2 for
both of the samples
78Mann-Whitney U Test
- Now compute the U statistic as follows
- U1 n1n2 (n1(n11)/2) T1
- U2 n1n2 (n2(n21)/2) T2
- Look up the appropriate a value in the table
given n2 - The Table will give you a value for Uo on the
left hand side corresponding to your n1 - Your computed U (smaller of U1 or U2) should be
less than the U stated in the table in order to
reject the Null hypothesis (that the population
relative frequency distributions are identical)
79Wilcoxon Signed Rank Test
- Similar to the Mann-Whitney U Test
- Allows you to compare paired differences
- Given n pairs of observations from populations A
and B, compute the paired differences (xA-xB) for
each pair of values - Rank the positive differences and the negative
differences separately - Compute the sums T and T- of these rankings
- For a one tailed test, use T- and for a two
tailed test, use the smaller of T or T- - Reject Ho if T lt To (critical value) obtained
from the Wilcoxon Table, given n and a values
80Other Nonparametric Tests
- Kruskal-Wallis H Test
- Just as the Mann-Whitney U Test is the
nonparametric alternative to the Students T-Test
for comparing population means, the
Kruskal-Wallis H Test is the nonparametric
alternative to ANOVA for a completely randomized
design and is used to detect differences in
location among more than 2 population
distributions based on independent random
sampling - Friedman Fr Test for Randomized Block Designs
- Is a nonparametric test for comparing the
distributions of measurements for k treatments
laid out in b blocks using a randomized block
design
81Test of Association
- Spearmans Rank Correlation Test
- Tests whether there is an association between 2
populations - Assume n pairs (xi, yi) of observations from 2
populations X, Y - Rank each of the xi and yi in ascending order
- Compute
- Rs Sxy / sqrt (Sxx Syy)
- Then given n and a, look up Ro in the Spearman
Table - Reject Ho (no association) if Rs gt Ro or Rs lt
-Ro
82Survival Analysis
83Introduction
- There are many clinical studies that address the
question of time to an event - For example, we often want to know given risk
factors, what is a patients chance for an MI?
(I.e. time to MI) - This type of data is called censored data
- Survival Analysis seeks to study this type of
question
84Life Tables
- The most straightforward way to compute a data
structure known as a Life Table - The entire lifetime of a study object is divided
into intervals of specified length - For each interval, the number of subjects
surviving or died within that interval is
determined and plotted - Based on this number, we can compute several
types of statistics - Numbers of cases at risk
- Proportion Failing or Proportion Surviving
- Probability Density or Hazard Rate
- Median Survival Time
- Required Sample Sizes
85Survival Analysis
- Although life tables give us a good estimate of
the risk of adverse events, it is desirable to
understand the underlying survival function
algorithmically for prediction purposes - The three distributions proposed for this are
the - Exponential (linear exponential) distribution
- Weibull Distribution
- Gompertz Distribution
- The parameter estimation procedure is then a
modified version of the least-squares model - And the statistic used to study it is an
incremental Chi-Square Statistic
86Kaplan-Meier Product Limit Estimator
- Rather than classify the survival into a
life-table, the KM estimator computes a survival
function directly from continuous survival or
failure times - Imagine creating a life table with exactly one
observation for each interval - Then we avoid the effect of grouping
observations together into interval categories - Then S(t) Productj ((n-j)/(n-j1))d(j)
- n total observations
- d(j) 1 if censored, 0 if not in interval j
87Comparing Survival Times
- Often we wish to compare survival times in 2 or
more populations - There are several tests available for this
purpose - Gehans Generalized Wilcoxon Test
- Cox-Mantel Test
- Coxs F-Test
- Log-Rank Test
- Peto and Petos Wilcoxon Test
- These are mostly nonparametric tests that
generate Z-values for comparing means
88Regression Models
- We also want to be able to predict survival time
given some independent risk factors - This is very common in the medical literature
- The regression test of choice is the
Cox-Proportional Hazards Model - The model is written as
- h(t), (z1, z2, ..., zm) h0(t)exp(b1z1
... bmzm) - (where h(t,...) denotes the resultant hazard,
given the values of the m covariates for the
respective case (z1, z2, ..., zm) and the
respective survival time (t). The term h0(t) is
called the baseline hazard it is the hazard for
the respective individual when all independent
variable values are equal to zero). We can
linearize this model by dividing both sides of
the equation by h0(t) and then taking the natural
logarithm of both sides - logh(t), (z...)/h0(t) b1z1 ...
bmzm - We now have a fairly "simple" linear model that
can be readily estimated)
89Useful Links
- http//hesweb1.med.virginia.edu/biostat/teaching/h
andouts.html - http//stat.tamu.edu/stat30x/notes/trydouble2.html
- http//www.statsoft.com/textbook/stathome.html
- http//davidmlane.com/hyperstat/index.html
- http//members.aol.com/johnp71/javastat.html
- http//www.helsinki.fi/jpuranen/links.html
- http//ubmail.ubalt.edu/harsham/statistics/REFSTA
T.HTMrgenRes - http//trochim.human.cornell.edu/kb/index.htm
90Questions