Medical Statistics: Hypothesis Testing

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Medical Statistics Hypothesis Testing
Nimrod Lavi, MD Adhir Shroff, MD, MPH
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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Continuous variable

• One in which research participants differ in
  degree or amount.
• susceptible to infinite gradations (p. 176,
  Pedhazur Schmelkin, 1991)
• Examples height, weight, age

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Categorical variable

• Participants belong to, or are assigned to,
  mutual exclusive groups
• Nominal
• Used to group subjects
• Numbers are arbitrary
• Examples sex, race, dead/alive, marital status
• Ordinal (rank)
• Given a numerical value in accordance to their
  rank on the variable
• Numerical values assigned to participants tells
  nothing of the distance between them
• Examples class rank, finishers in a race

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Independent vs Dependent Variable

• Independent
• predictor variable
• Usually on the x axis
• Dependent
• outcome variable
• Usually on the y axis

• The independent variable (a treatment) leads to
  the dependent variable (outcome)
• Ultimately, we are interested in differences
  between dependent variables

Dependent
Independent
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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Descriptive Statistics

• These are measures or variables that summarize a
  data set
• 2 main questions
• Index of central tendency (ie. mean)
• Index of dispersion (ie. std deviation)

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Descriptive Statistics

• Data set for ICD complications in 2005
• 14 patients
• Sex F, F, M, M, F, F, F, M, F, M, M, F, F, F
• Make G, S, G, G, G, M, S,S, G,G, M, S

• Central tendency is summarized by proportion or
  frequency
• Sex
• M 5/14 .36 or 36
• F 9/14 .64 or 64
• Make
• G 6/12 .5 or 50
• S 4/12 .33 or 33
• M 2/12 .17 or 17
• Dispersion not really used in categorical data

• Categorical data

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Descriptive Statistics

• Data set SBP among a group of CHF pts in VA
  clinic
• 13 patients
• 100, 95, 98, 172, 74, 103, 97, 106, 100, 110,
  118, 91, 108

• Central Tendency
• Mean
• mathematical average of all the values
• S (xixiixn)/n
• Median
• value that occupies middle rank, when values are
  ordered from least to greatest
• Mode
• Most commonly observed value(s)

• Continuous variable

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Descriptive Statistics

• Data set SBP among a group of CHF pts in VA
  clinic
• 13 patients
• 100, 95, 98, 172, 74, 103, 97, 106, 100, 110,
  118, 91, 108

• Central Tendency
• Mean
• mathematical average of all the values
• S (xixiixn)/n
• (100959817274103 97106100110118
  91108)/13 105.5

• Continuous variable

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Descriptive Statistics

• Data set SBP among a group of CHF pts in VA
  clinic
• 13 patients
• 100, 95, 98, 172, 74, 103, 97, 106, 100, 110,
  118, 91, 108

• Central Tendency
• Median
• value that occupies middle rank, when values are
  ordered from least to greatest
• 74, 91, 95, 97, 98, 100, 100,
• 103, 106, 108, 110, 118,
• 172
• Useful if data is skewed or there are outliers

• Continuous variable

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Descriptive Statistics

• Data set SBP among a group of CHF pts in VA
  clinic
• 100, 95, 98, 172, 74, 103, 97, 106, 100, 110,
  118, 91, 108

• Index of dispersion
• Standard deviation
• measure of spread around the mean
• Calculated by measuring the distance of each
  value from the mean, squaring these results (to
  account for negative values), add them up and
  take the sq root

• Continuous variable

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Descriptive Statistics Normal
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Descriptive Statistics Confidence Intervals

• Range of values which we can be confident
  includes the true value
• Defines the inner zone about the central index
  (mean, proportion or ration)
• Describes variability in the sample from the mean
  or center
• Will find CI used in describing the difference
  between means or proportions when doing
  comparisons between groups

Altman DG. Practical Statistics for Medical
Research 1999
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Descriptive Statistics Confidence Intervals

• For example, a 95 CI indicates that we are 95
  confident that the population mean will fall
  within the range described
• Can be used similar to a p-value to determine
  significant differences
• CI is similar to a measure of spread, like SD
• As sample size increase or variability in the
  measurement decrease, the CI will become more
  narrow

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Descriptive Statistics Confidence Intervals
L a n c e t 1999 3 5 4 7 0 8 1 5

• Prospective, randomized, multicenter trial of
  different management strategies for ACS
• 2500 pts enrolled in Europe with 6 month
  follow-up
• Primary endpoints Composite endpoint of death
  and myocardial infarction after 6 months

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Descriptive Statistics Confidence Intervals
L a n c e t 1999 3 5 4 7 0 8 1 5
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Descriptive Statistics Confidence Intervals
L a n c e t 1999 3 5 4 7 0 8 1 5
Risk ratio Riskinvasive / Risknoninvasive
When CI cross 1 or whatever designates
equivalency, the p-value not be significant.
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Descriptive Statistics Confidence Intervals
L a n c e t 1999 3 5 4 7 0 8 1 5

• Review
• Calculate
• RRR, ARR, NNT

• RRR (12.1-9.4) / 12.1 22

ARR 12.1 - 9.4 2.7
NNT 100 / ARR 100 / 2.7 37
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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Hypothesis

• Statement about a population, where a certain
  parameter takes a particular numerical value or
  falls in a certain range of values.
• Examples
• A director of an HMO hypothesizes that LOS p AMI
  is longer than for CHF exacerbation
• An investigator states that a new therapy is 10
  better than the current therapy
• Bivalirudin is not-inferior to heparin/eptifibitid
  e for coronary PCI

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Null Hypothesis (Ho)

• Innocent until proven guilty
• Null hypothesis (Ho) usually states that no
  difference between test groups really exists
• Fundamental concept in research is the concept of
  either rejecting or conceding the Ho
• State the Ho

• A director of an HMO hypothesizes that LOS p AMI
  is longer than for CHF exacerbation
• An investigator states that a new therapy is 10
  better than the current therapy
• Bivalirudin is not-inferior to heparin/eptifibitid
  e for PCI

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Null Hypothesis (Ho) Courtroom Analogy

• The null hypothesis is that the defendant is
  innocent.
• The alternative is that the defendant is guilty.
• If the jury acquits the defendant, this does not
  mean that it accepts the defendants claim of
  innocence.
• It merely means that innocence is plausible
  because guilt has not been established beyond a
  reasonable doubt.

Graduate Workshop in Statistics Session 4.
Hamidieh K. 2006 Univ of Michigan
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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Extrapolation of Research Findings

• Sample population vs. the world
• If your study shows that treatment A is better
  than treatment B
• You cannot conclude that treatment A is ALWAYS
  better than treatment B
• You only sampled a small portion of the entire
  population, so there is always a chance that your
  observation was a chance event

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Extrapolation of Research Findings

• At what point are we comfortable concluding that
  there is a difference between the groups in our
  sample
• In other words, what is the false-positive rate
  that we are willing to accept
• What is this called in statistical terms?

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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Definition of p-value

• With any research study, there is a possibility
  that the observed differences were a chance event
• The only way to know that a difference is really
  present with certainty, the entire population
  would need to be studied
• The research community and statisticians had to
  pick a level of uncertainty at which they could
  live

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Definition of p-value

• This level of uncertainty is called type 1 error
  or a false-positive rate

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Two Types of Errors
Trt has no effect
Trt has an effect
Graduate Workshop in Statistics Session 4.
Hamidieh K. 2006 Univ of Michigan
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Definition of p-value

• This level of uncertainty is called type 1 error
  or a false-positive rate (a)
• More commonly called a p-value
• Statistical significance will be recognized if
  p 0.05 (can be set lower if one wishes)

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Trade-Off in Probability for Two Errors
There is an inverse relationship between the
probabilities of the two types of
errors. Increase probability of a type I error
? decrease in probability of a type II error
.05
.01
Graduate Workshop in Statistics Session 4.
Hamidieh K. 2006 Univ of Michigan
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Definition of p-value

• This level of uncertainty is called type 1 error
  or a false-positive rate (a)
• More commonly called a p-value
• In general, p 0.05 is the agreed upon level
• In other words, the probability that the
  difference that we observed in our sample
  occurred by chance is less than 5
• Therefore we can reject the Ho

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Definition of p-value
Stating the Conclusions of our Results

• When the p-value is small, we reject the null
  hypothesis or, equivalently, we accept the
  alternative hypothesis.
• Small is defined as a p-value ? a, where a
  acceptable false () rate (usually 0.05).
• When the p-value is not small, we conclude that
  we cannot reject the null hypothesis or,
  equivalently, there is not enough evidence to
  reject the null hypothesis.
• Not small is defined as a p-value gt a, where a
  acceptable false () rate (usually 0.05).

Graduate Workshop in Statistics Session 4.
Hamidieh K. 2006 Univ of Michigan
36
Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• t-tests
• Chi-square

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Two Sample Tests Continuous Variable

• t-test
• Comparing two groups, statistical significance is
  determined by
• Magnitude of the observed difference
• Bigger differences are more likely to be
  significant
• Spread, or variability, of the data
• Larger spread will make the differences not
  significant

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Two Sample Tests Continuous Variable
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Two Sample Tests Continuous Variable

• t-test
• Comparing two groups, statistical significance is
  determined by
• Magnitude of the observed difference
• Bigger differences are more likely to be
  significant
• Spread, or variability, of the data
• Larger spread will make the differences not be
  significant
• Key is to compare the difference between groups
  with the variability within each group

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Two Sample Tests Continuous Variable

• Types t-tests
• Student t-test or two sample t-test
• Used if independent variables are unpaired
• Example
• A randomized trial to high dose statin versus
  placebo post AMI
• Paired t-test
• Used if independent variables are paired
• Each person is measured twice under different
  conditions
• Similar individuals are paired prior to an
  experiment
• Each receives a different trt, same response is
  measured
• Example
• A study of ejection fraction in patients before
  and after Bi-V pacing

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Two Sample Tests Continuous Variable

• t-test
• Tails
• Two-tailed
• Most commonly used in clinical research studies
• Means that the treatment group can be better or
  worse than the control group
• One-tailed
• Used only if the groups can only differ in one
  direction

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Example t-test

• What type of test should be run?
• How are the data related or are they?

• Data entered into a statistical program
• p value 0.2329, not significant

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Agenda

• Types of variables
• Descriptive statistics
• What is a hypothesis
• Definition of a p-value
• Sample vs. universe
• Comparative statistics
• T-tests
• Chi-square

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Two Sample Tests Categorical Variables

• Chi square (?2) analysis
• Data that is organized into frequency, generate
  proportions
• Based on comparing what values are expected from
  the null hypothesis to what is actually observed
• Greater the difference between the observed and
  expected, the more likely the result will be
  significant

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Chi square (?2) analysis
Outcome
Therapy
Totals
abcd

• Null hypothesis states that outcomes of therapy
  A and B are equally successful
• This is how the expected outcomes are determined

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Chi square (?2) analysis
Outcome
Therapy
Totals
abcd

• Next the actual observed values are then
  recorded
• With this information the ?2 value can be
  calculated and a p-value will be generated

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Example ?2 analysis

• Arrange data into a 2x2 table
• Treatment groups along the vertical axis,
  Outcomes alone the horizontal axis

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Example ?2 analysis

• Data entered into a statistical program

• P-value 0.6392
• Not a significant difference

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Example Ear Infections and Xylitol
Experiment n 533 children randomized to 3
groups Group 1 Placebo Gum Group 2
Xylitol Gum Group 3 Xylitol
Lozenge Response Did child have an ear
infection?
Group Infection Count 1 placebo Y
49 2 gum N 150 3 lozenge
Y 39 4 placebo N 129 5 gum
Y 29 6 lozenge N 137
Graduate Workshop in Statistics Session 5.
Hamidieh K. 2006 Univ of Michigan
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Two Sample Tests Categorical Variables
Outcome
Therapy
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Example Ear Infections and Xylitol
Compute expected count for each cell Expected
count (Row total) ? (Column total) / Total n
Example 39.1 (178 117) / 533 Or
intuitively, calculate overall infection rate
total number infected / total number
117/533 .2195 Now, assuming no difference
between treatments, the infection rate will be
the same in each group .2195 x total for each
group .2195 x 178 39.1
Graduate Workshop in Statistics Session 5.
Hamidieh K. 2006 Univ of Michigan
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Example Ear Infections and Xylitol
? From a table, p 0.035
Graduate Workshop in Statistics Session 5.
Hamidieh K. 2006 Univ of Michigan
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Conclusion

• There are many ways to describe ones data
• P-values are the maximum acceptable false
  positive rate
• Remember the Courtroom Analogy when it comes to
  the Null hypothesis
• Choice of statistical test depends on type of
  variable and number of comparison groups

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References

• Neely JG, et al.
• Laryngoscope, 11212491255, 2002
• Laryngoscope, 11315341540, 2003
• Laryngoscope, 1131719 1724, 2003
• Guyatt G, et al. Basic Statistics for Clinicians.
  CMAJ. 1/1/95
• http//www-personal.umich.edu/khamidie/?MA
• Altman, DG. Practical Statistics for Medical
  Research. 1999.

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• Thank you