Fundamental Concepts of Biostatistics

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Fundamental Concepts of Biostatistics

• Cathy Jenkins, MS
• Biostatistician II
• Lisa Kaltenbach, MS
• Biostatistician II
• April 17, 2007

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Prior to any analysis

• Define research question(s). Write out using no
  more than one sentence per question.
• Determine statistical analysis plan to address
  each research question.
• Analysis

Confounder
Outcome
Predictor
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Population versus Sample

• Population includes all possible observations of
  a particular type.
• Observations may be people, animals, places, or
  things
• Ex. Men women aged 18 and older infants
  penguins bodies of water
• Sample includes only some of the observations
  but selected in a way that gives every possible
  observation an equal chance of being observed.
• Ex. Men women aged 18 and older in the
  TennCare Database from years 1998-2005 infants
  with a primary care physician at Vanderbilt
  during years 1990-2000 all penguins living in
  the Nashville zoo since 1995 bodies of water
  included in Bay Delta Tributaries Database

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Population versus Sample 2

• In most cases in clinical research, we want to
  generalize from information about our sample to
  information about a population.

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Descriptive Statistics

• To describe characteristics of the sample
• Ex. demographics, distributions, frequencies
• May want to describe data with numerical or
  graphical summary
• Characteristics of sample may be continuous or
  categorical variables

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Continuous Variables

• Continuous a variable that can take on any
  number of possible values (Ex. weight).
• Discrete Numeric a variable whose set of
  possible values is a finite sequence of numbers
  (Ex. pain scale 1 to 5).
• Numerical Summary
• Often want to measure central tendency of data
• Sample mean The sum of all of the observations
  divided by the number of observations. The mean
  is only useful when the data are normally
  distributed.
• Sample Median (50th Percentile) Order the
  observations from smallest to largest
• If n is odd, then the median is the middle
  ordered observation.
• If n is even, then the median is the average of
  the two middle ordered observations.

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Continuous Variables 2

• Other common percentiles include quartiles (25th,
  50th, and 75th percentiles) and deciles (10th,
  20th, , 90th percentiles).
• The p-th percentile is the value that p- of the
  data are less than or equal to. If p- of the
  data lie below the p-th percentile, it follows
  the (100- p)- of the data lie above it.
• Ex If the 85-th percentile of household income
  is 60,000 then 85 of the households have
  incomes of 60,000 or less and the top 15 of
  households have incomes of 60,000 or more.
• Measures of Dispersion
• When measurements are collected there will be
  scatter, dispersion, or variability.
• Sources of dispersion
• Random error Error due to chance
• Systematic Error Wrong result do to bias
• Biological variability

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Continuous Variables 3

• Minimum Smallest observed value
• Maximum Largest observed value
• Range Difference between max and min (often
  reported as (min, max))
• Interquartile Range (IQR) Difference between
  75th and 25th percentiles (often reported as
  (25th,75th))
• Variance The average of the squares of the
  deviations of the observations from their mean.
• Standard Deviation the square root of the
  variance
• Standard deviation measures spread about the mean
  and should be used only when the mean is chosen
  as the measure of center
• Has the same unit of measurement as the mean

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Continuous Variables 4

• Graphical Summary

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Categorical Variables

• Categorical a variable having only certain
  possible values (ex. race).
• Binary a categorical variable with only two
  possible values (ex. gender).
• Ordinal a categorical variable for which there
  is a definite ordering of the categories (ex.
  severity of lower back pain ordered as none,
  mild, moderate, and severe).
• Numerical Summary
• Frequency Distribution A listing of distinct
  values for that characteristic and the number of
  observations having each value.
• Relative frequency Proportion of the total
  number of observations that fall into each
  category.
• Cumulative frequency Proportion of the total
  number of observations that fall into the current
  or previous categories listed (may be useful for
  ordinal variable).

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Relationships between two variables

• Two variables measured on the same observations
  are associated if some values of the first
  variable tend to occur more often with some
  values of the second variable than with other
  values of that variable.
• Two continuous variables
• Ex. Persons weight and blood pressure
• Two categorical variables
• Ex. gender and smoking status
• One continuous one categorical variable
• Ex. blood pressure and gender
• Keep in mind - relationship between two variables
  can be strongly influenced by other variables
  that are lurking in the background

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Two Continuous Variables

• A scatterplot shows the relationship between two
  continuous variables measured on the same
  observations.
• The values of one variable appear on the x-axis,
  and the values of the other variable appear on
  the y-axis.
• Each observation appears as a point in the plot
  fixed by the values of both variables for that
  observation.

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Two Continuous Variables 2

• Graphical Summary Scatterplot

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Two Categorical Variables

• Numerical Summary
• Cross-tabulation (or 2-way table)
• Ex. Clinical Pregnancy by Age Groups

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One Categorical and One Continuous variable

• Consider descriptive statistics of the continuous
  variable separately for different values of the
  categorical variable
• Ex Descriptive statistics of birth weight by
  smoking status during pregnancy for mothers

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How to study the relationship between two
different variables

• Quantify the relationship Measure the strength
  of the relationship (linear, monotonic, )
  between two continuous variables.
• Use hypothesis testing Test theory to see if
  experimental results only reflect random chance.
• Fit model Predict one measure of an individual
  from another.

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Quantify the relationship

• Correlation coefficient (r) Quantitative
  summary of the strength of the relationship
  between two continuous variables.
• Pearson correlation focuses on the raw data.
• Spearman correlation focuses on the ranks of
  the raw data.
• Covariance (r2) Square of the correlation
  coefficient that defines
• the strength or magnitude of the correlation.
• not a cause and effect relationship but
  quantifies how well one variable predicts
  another.

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Hypothesis testing 1

• Define null hypothesis for question of interest
  that assumes the experimental results are due to
  chance alone.
• Perform statistical test to determine if we can
  reject or fail to reject the null hypothesis.
• NOTE Absence of evidence does not mean evidence
  of absence. In other words, if our test results
  in a non-significant p-value, we do not accept
  the null hypothesis. Rather we fail to reject
  the null hypothesis. It could be that for the
  same experiment but a different sample we would
  obtain significant results.
• P-value the probability of obtaining a result
  at least as extreme as a given data point
  assuming the data point was the result of chance
  alone.

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Hypothesis testing 2

• Categorical data
• Chi-square tests
• Both row and column variables are nominal
• Row variable nominal column variable ordinal
• Both row and column variables are ordinal
• Tests whether distribution of frequencies differs
  across rows (groups) or whether there is any
  association between the row and column variables.

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Hypothesis testing 3

• Nominal row and column variables
• Example Given data on the neighborhood in which
  a person lives and his political affiliation, you
  wish to test whether a persons politics
  influences where he/she lives.
• H0 No association exists between a persons
  political affiliation and the neighborhood in
  which he lives.
• HA An association exists between a persons
  political affiliation and the neighborhood in
  which he lives.

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Hypothesis testing 4

• Nominal row variables and ordinal column
  variables
• Example Given data studying hours of headache
  pain relief (hours ranging from 0 6) using
  three different treatments placebo, standard,
  and test treatment.
• H0 No association between hours of pain relief
  and treatment.
• HA A shift in row mean hours of headache pain
  relief exists between the treatment groups.

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Hypothesis testing 5

• Ordinal row and column variables
• Example Given data assessing how water
  additives (water, standard, super) affect the
  washability of clothes (low, medium, high).
• H0 No association between the water additive
  and the washability of the clothes.
• HA There is a linear association between water
  additive and washability of clothes.

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Hypothesis Testing 6

• Continuous variables
• Parametric tests Make assumptions about
  underlying distribution of data.
• 1-sample t-test H0 Mean of data is equal to
  some fixed value (defined by study question).
• 2-sample t-test H0 No difference in means
  between the two independent groups.
• Paired t-test H0 Mean of difference in paired
  data is equal to 0.

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Hypothesis testing 7

• Non-parametric tests No assumptions about
  underlying distribution of data.
• Wilcoxon signed rank test analogous to the
  one-sample or paired t-test.
• 1-sample H0 Median is equal to specified value
  (defined by study question).
• Paired H0 Median difference is equal to 0.
• Wilcoxon rank sum test analogous to the
  two-sample t-test.
• H0 The distribution of the response variable is
  the same in the two independent groups.

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Modeling 1 ANOVA

• 1-way/2-way extends 2-sample t-test (with 1
  factor/2 factors) to n-groups -- compares mean of
  continuous variable across n-groups.
• H0 No difference in means between the n-groups.
• HA At least one group has a different mean than
  the other (n-1) groups.
• Avoids problems with multiple comparisons.
• Tests whether within-group variability is greater
  than between-group variability.

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Modeling 2 Linear regression

• Continuous outcome.
• Assumes relationship between predictor(s) and
  outcome is linear.
• Observations assumed to be independent (ie., only
  one observation per subject, no subjects that are
  related to each other, etc.)
• Number of predictors allowed in the model depends
  on the sample size.
• Rule of thumb no more than n/10 predictors where
  n of subjects.
• Include confounders in the model for better
  parameter estimates.
• Output are parameter estimates
• Can give information similar to that obtained
  from hypothesis testing.
• Allows the investigator to make inference based
  on the parameter estimates.

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Modeling 3Logistic regression

• Categorical outcome typically binary.
• Number of predictors in model depends on several
  things
• Each group has at least 10 subjects.
• Cell counts in a cross-tab table meet certain
  sample size criteria
• 80 of expected counts are at least 5.
• All other expected counts are greater than 2,
  with virtually no 0 counts.
• Output are parameter estimates and odds ratios
  calculated from these parameter estimates.
• Odds ratio way of comparing whether the
  probability of a certain event is the same for
  two groups.
• OR 1 ? The event is equally likely in both
  groups.
• OR gt 1 ? The event is more likely in the first
  group.
• OR lt 1 ? The event is less likely in the first
  group.

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Conclusion

• Statistical analysis plan should be devised
  before collecting data.
• Use aims of study to decide the best way to study
  the relationship between variables of interest
  correlations, hypothesis testing, modeling.
• Make use of the daily biostatistics clinics.
  Refer to http//biostat.mc.vanderbilt.edu for the
  clinic schedule. Click on the Clinics link
  (5th link from the top).
• Check to see if your department is a part of the
  collaboration plan for more intensive
  biostatistics support.