Introduction to Clinical Biostatistics for Medical Students

1
Introduction to Clinical Biostatistics for
Medical Students

• Atif Zafar, MD
• Department of Medicine

2
Overview of Presentation

• Introductory Concepts (Review)
• Hypothesis Testing
• Linear Regression and Correlation
• Analysis of Variance (ANOVA)
• Nonparametric Statistics
• Survival Analysis

3
Introductory Concepts
4
Introductory Concepts

• Types of Data
• Presenting Data
• Descriptive Measures
• Probability and Distributions
• Estimation Techniques

5
Types 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

6
Types 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)

7
Types 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.)

8
Types 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

9
Types 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

10
Types 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

11
Presenting 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

12
Presenting Data

• Scatter Plots (Plot of a Bivariate experiment)

13
Presenting 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

14
Presenting 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

15
Presenting Data

• Boxplots

16
Descriptive 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

17
Measures 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

18
Measures of Variability

• Once we have located the center of a set of data
  points, we want to know how dispersed they are

19
Measures 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

20
Measures 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

21
Measures 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

22
Probability

• 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

23
Probabilities 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

24
Probability 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

25
Probability 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

26
Probability Distributions

• Normal Distribution (N)
• Applies to continuous random variables
• Standard Normal Distribution (Z)
• A Normal Distribution with
• Mean of 0
• Standard Deviation of 1

27
Estimation 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

28
Estimation 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

29
Estimation 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

30
Estimation 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

31
Hypothesis Testing

• Large Sample Estimation Techniques

32
Introduction 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.

33
Confidence 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

34
Significance 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

35
P-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

36
Comparing 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)

37
Hypothesis 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

38
Hypothesis 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

39
Constructing 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)

40
Testing 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

41
Confidence 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)

42
Types 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

43
Types 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

44
T-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.

45
Chi-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)

46
F-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

47
Linear Regression and Correlation
48
Linear 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

49
Linear 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

50
Linear 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

51
Method 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
52
Inferences 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

53
Inferences 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

54
Inferences 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

55
Linear Regression

• So, graphically what does it look like?

56
Other 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

57
Correlation

• 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

58
Correlation

• 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

59
Multiple 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

60
Multiple 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

61
Multiple 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

62
Multiple 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

63
Logistic 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.

64
The ANalysis Of VAriance

• Also known as ANOVA

65
ANOVA

• 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

66
ANOVA

• 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

67
ANOVA

• 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

68
ANOVA 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

69
ANOVA 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

70
ANOVA 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

71
ANOVA 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)

72
ANOVA 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

73
Nonparametric Statistics

• Analysis of Ranked Data

74
Nonparametric 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)

75
Nonparametric 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

76
The 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

77
Mann-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

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Mann-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)

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Wilcoxon 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

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Other 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

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Test 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

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Survival Analysis
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Introduction

• 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

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Life 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

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Survival 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

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Kaplan-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

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Comparing 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

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Regression 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)

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Useful 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

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Questions

• Thank You