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Statistical Methods in Clinical Research
James B. Spies M.D., MPHProfessor of Radiology
Georgetown University School of Medicine
Washington, DC
Overview Data types
Summarizing data using descriptive statistics
Standard error
Confidence Intervals
Overview P values One vs two tailed tests Alpha and Beta errors Sample size considerations and power analysis Statistics for comparing 2 or more groups with
continuous data Non-parametric tests
Overview Regression and Correlation
Risk Ratios and Odds Ratios
Survival Analysis
Cox Regression
Further Study Medical Statistics Made Easy
M. Harris and G. Taylor Informa Healthcare UK
Distributed in US by:Taylor and Francis
6000 Broken Sound Parkway, NW Suite 300
Boca Raton, FL 33487
1-800-272-7737
Types of Data Discrete Data-limited number of choices
Binary: two choices (yes/no) Dead or alive Disease-free or not
Categorical: more than two choices, not ordered Race Age group
Ordinal: more than two choices, ordered Stages of a cancer Likert scale for response
E.G. strongly agree, agree, neither agree or disagree, etc.
Types of data Continuous data
Theoretically infinite possible values (within physiologic limits) , including fractional values
Height, age, weight Can be interval
Interval between measures has meaning. Ratio of two interval data points has no meaning Temperature in celsius, day of the year).
Can be ratio Ratio of the measures has meaning Weight, height
Types of Data Why important? The type of data defines:
The summary measures used Mean, Standard deviation for continuous data Proportions for discrete data
Statistics used for analysis: Examples:
T-test for normally distributed continuous Wilcoxon Rank Sum for non-normally distributed
continuous
Descriptive Statistics Characterize data set
Graphical presentation Histograms Frequency distribution Box and whiskers plot
Numeric description Mean, median, SD, interquartile range
HistogramContinuous Data
No segmentation of data into groups
Frequency Distribution
Segmentation of data into groupsDiscrete or continuous data
Box and Whiskers Plots
Box and Whisker Plots
Popular in Epidemiologic StudiesUseful for presenting comparative data graphically
Numeric Descriptive Statistics Measures of central tendency of data
Mean Median Mode
Measures of variability of data Standard Deviation Interquartile range
Sample Mean Most commonly used measure of central tendency
Best applied in normally distributed continuous data.
Not applicable in categorical data
Definition: Sum of all the values in a sample, divided by the number of
values.
Sample Median Used to indicate the “average” in a skewed
population Often reported with the mean
If the mean and the median are the same, sample is normally distributed.
It is the middle value from an ordered listing of the values
If an odd number of values, it is the middle value If even number of values, it is the average of the two middle
values. Mid-value in interquartile range
Sample Mode Infrequently reported as a value in studies.
Is the most common value
More frequently used to describe the distribution of data Uni-modal, bi-modal, etc.
Interquartile range Is the range of data from the 25th percentile
to the 75th percentile
Common component of a box and whiskers plot It is the box, and the line across the box is the
median or middle value Rarely, mean will also be displayed.
Standard Error A fundamental goal of statistical analysis is to
estimate a parameter of a population based on a sample
The values of a specific variable from a sample are an estimate of the entire population of individuals who might have been eligible for the study.
A measure of the precision of a sample in estimating the population parameter.
Standard Error Standard error of the mean
Standard deviation / square root of (sample size) (if sample greater than 60)
Standard error of the proportion Square root of (proportion X 1 - proportion) / n)
Important: dependent on sample size Larger the sample, the smaller the standard error.
Clarification Standard Deviation measures the
variability or spread of the data in an individual sample.
Standard error measures the precision of the estimate of a population parameter provided by the sample mean or proportion.
Standard Error Significance:
Is the basis of confidence intervals
A 95% confidence interval is defined by Sample mean (or proportion) ± 1.96 X standard error
Since standard error is inversely related to the sample size:
The larger the study (sample size), the smaller the confidence intervals and the greater the precision of the estimate.
Confidence Intervals May be used to assess a single point
estimate such as mean or proportion.
Most commonly used in assessing the estimate of the difference between two groups.
Confidence Intervals
Commonly reported in studies to provide an estimate of the precisionof the mean.
Confidence Intervals
P Values The probability that any observation is due to chance
alone assuming that the null hypothesis is true Typically, an estimate that has a p value of 0.05 or less is
considered to be “statistically significant” or unlikely to occur due to chance alone.
The P value used is an arbitrary value P value of 0.05 equals 1 in 20 chance P value of 0.01 equals 1 in 100 chance P value of 0.001 equals 1 in 1000 chance.
P Values and Confidence Intervals P values provide less information than confidence
intervals. A P value provides only a probability that estimate is due to chance A P value could be statistically significant but of limited clinical
significance. A very large study might find that a difference of .1 on a VAS Scale of 0
to 10 is statistically significant but it may be of no clinical significance A large study might find many “significant” findings during
multivariable analyses.
“a large study dooms you to statistical significance”
Anonymous Statistician
P Values and Confidence Intervals Confidence intervals provide a range of plausible values of the
population mean For most tests, if the confidence interval includes 0, then it is not
significant. Ratios: if CI includes 1, then is not significant
The interval contains the true population value 95% of the time. If a confidence interval range is very wide, then plausible value
might range from very low to very high. Example: A relative risk of 4 might have a confidence interval of 1.05 to
9, suggesting that although the estimate is for a 400% increased risk, an increased risk of 5% to 900% is plausible.
Errors Type I error
Claiming a difference between two samples when in fact there is none.
Remember there is variability among samples- they might seem to come from different populations but they may not.
Also called the error. Typically 0.05 is used
Errors Type II error
Claiming there is no difference between two samples when in fact there is.
Also called a error. The probability of not making a Type II
error is 1 - , which is called the power of the test.
Hidden error because can’t be detected without a proper power analysis
Errors
Null Hypothesis
H0
Alternative Hypothesis
H1
Null Hypothesis
H0
No Error Type I
Alternative Hypothesis
H1
Type II
No Error
Test Result
Truth
Sample Size Calculation Also called “power analysis”. When designing a study, one needs to determine
how large a study is needed. Power is the ability of a study to avoid a Type II error. Sample size calculation yields the number of study
subjects needed, given a certain desired power to detect a difference and a certain level of P value that will be considered significant.
Many studies are completed without proper estimate of appropriate study size.
This may lead to a “negative” study outcome in error.
Sample Size Calculation
Depends on: Level of Type I error: 0.05 typical Level of Type II error: 0.20 typical One sided vs two sided: nearly always two Inherent variability of population
Usually estimated from preliminary data The difference that would be meaningful
between the two assessment arms.
One-sided vs. Two-sided Most tests should be framed as a two-
sided test. When comparing two samples, we usually
cannot be sure which is going to be be better.
You never know which directions study results will go.
For routine medical research, use only two-sided tests.
Sample size for proportions
Stata input: Mean 1 = .2, mean 2 = .3, = .05, power (1-) =.8.
Sample Size for Continuous Data
Stata input: Mean 1 = 20, mean 2 = 30, = .05, power (1-) =.8, std. dev. 10.
Statistical Tests Parametric tests
Continuous data normally distributed
Non-parametric tests Continuous data not normally distributed Categorical or Ordinal data
Comparison of 2 Sample Means Student’s T test
Assumes normally distributed continuous data.
T value = difference between means standard error of difference
T value then looked up in Table to determine significance
Paired T Tests Uses the change before
and after intervention in a single individual
Reduces the degree of variability between the groups
Given the same number of patients, has greater power to detect a difference between groups
Analysis of Variance Used to determine if two or more samples are
from the same population- the null hypothesis. If two samples, is the same as the T test. Usually used for 3 or more samples.
If it appears they are not from same population, can’t tell which sample is different. Would need to do pair-wise tests.
Non-parametric Tests Testing proportions
(Pearson’s) Chi-Squared (2) Test Fisher’s Exact Test
Testing ordinal variables Mann Whiney “U” Test Kruskal-Wallis One-way ANOVA
Testing Ordinal Paired Variables Sign Test Wilcoxon Rank Sum Test
Use of non-parametric tests Use for categorical, ordinal or non-normally
distributed continuous data May check both parametric and non-
parametric tests to check for congruity Most non-parametric tests are based on
ranks or other non- value related methods Interpretation:
Is the P value significant?
(Pearson’s) Chi-Squared (2) Test
Used to compare observed proportions of an event compared to expected.
Used with nominal data (better/ worse; dead/alive)
If there is a substantial difference between observed and expected, then it is likely that the null hypothesis is rejected.
Often presented graphically as a 2 X 2 Table
Chi-Squared (2) Test Chi-Squared (2) Formula
Not applicable in small samples If fewer than 5 observations per cell, use
Fisher’s exact test
Correlation Assesses the linear relationship between two variables
Example: height and weight Strength of the association is described by a correlation
coefficient- r r = 0 - .2 low, probably meaningless r = .2 - .4 low, possible importance r = .4 - .6 moderate correlation r = .6 - .8 high correlation r = .8 - 1 very high correlation
Can be positive or negative Pearson’s, Spearman correlation coefficient Tells nothing about causation
Correlation
Source: Harris and Taylor. Medical Statistics Made Easy
Correlation
Perfect Correlation
Source: Altman. Practical Statistics for Medical Research
Correlation
Source: Altman. Practical Statistics for Medical Research
Correlation Coefficient 0 Correlation Coefficient .3
Correlation
Source: Altman. Practical Statistics for Medical Research
Correlation Coefficient -.5 Correlation Coefficient .7
Regression Based on fitting a line to data
Provides a regression coefficient, which is the slope of the line
Y = ax + b Use to predict a dependent variable’s value based on the
value of an independent variable. Very helpful- In analysis of height and weight, for a known
height, one can predict weight. Much more useful than correlation
Allows prediction of values of Y rather than just whether there is a relationship between two variable.
Regression Types of regression
Linear- uses continuous data to predict continuous data outcome
Logistic- uses continuous data to predict probability of a dichotomous outcome
Poisson regression- time between rare events. Cox proportional hazards regression- survival
analysis.
Multiple Regression Models Determining the association between two
variables while controlling for the values of others.
Example: Uterine Fibroids Both age and race impact the incidence of fibroids. Multiple regression allows one to test the impact of
age on the incidence while controlling for race (and all other factors)
Multiple Regression Models In published papers, the multivariable models are
more powerful than univariable models and take precedence.
Therefore we discount the univariable model as it does not control for confounding variables.
Eg: Coronary disease is potentially affected by age, HTN, smoking status, gender and many other factors.
If assessing whether height is a factor: If it is significant on univariable analysis, but not on
multivariable analysis, these other factors confounded the analysis.
Risk Ratios Risk is the probability that an event will happen.
Number of events divided by the number of people at risk.
Risks are compared by creating a ratio Example: risk of colon cancer in those exposed to a factor
vs those unexposed Risk of colon cancer in exposed divided by the risk in those
unexposed.
Risk Ratios Typically used in cohort studies
Prospective observational studies comparing groups with various exposures.
Allows exploration of the probability that certain factors are associated with outcomes of interest For example: association of smoking with lung
cancer Usually require large and long-term studies
to determine risks and risk ratios.
Interpreting Risk Ratios A risk ratio of 1 equals no increased risk
A risk ratio of greater than 1 indicates increased risk
A risk ratio of less than 1 indicates decreased risk
95% confidence intervals are usually presented Must not include 1 for the estimate to be statistically
significant. Example: Risk ratio of 3.1 (95% CI 0.97- 9.41) includes 1, thus
would not be statistically significant.
Odds Ratios Odds of an event occurring divided by
the odds of the event not occurring. Odds are calculated by the number of
times an event happens by the number of times it does not happen.
Odds of heads vs the odds of tails is 1:1 or 1.
Odds Ratios Are calculated from case control studies
Case control: patients with a condition (often rare) are compared to a group of selected controls for exposure to one or more potential etiologic factors.
Cannot calculate risk from these studies as that requires the observation of the natural occurrence of an event over time in exposed and unexposed patients (prospective cohort study).
Instead we can calculate the odds for each group.
Comparing Risk and Odds Ratios For rare events, ratios very similar
If 5 of 100 people have a complication: The odds are 5/95 or .0526. The risk is 5/100 or .05.
If more common events, ratios begin to differ If 30 of 100 people have a complication:
The odds are 30/70 or .43 The risk is 30/100 or .30
Very common events, ratios very different Male versus female births
The odds are .5/.5 or 1 The risk is .5/1 or .5
Risk reduction Absolute risk reduction: amount that the risk is
reduced. Relative risk reduction: proportion or percentage
reduction. Example:
Death rate without treatment: 10 per 1000 Death rate with treatment: 5 per 1000 ARR = 5 per 1000 RRR = 50%
Survivial Analysis Evaluation of time to an event (death,
recurrence, recover). Provides means of handling censored data
Patients who do not reach the event by the end of the study or who are lost to follow-up
Most common type is Kaplan-Meier analysis Curves presented as stepwise change from
baseline There are no fixed intervals of follow-up- survival
proportion recalculated after each event.
Survival Analysis
Source: Altman. Practical Statistics for Medical Research
Kaplan-Meier Curve
Source: Wikipedia
Kaplan-Meier Analysis Provides a graphical means of comparing the
outcomes of two groups that vary by intervention or other factor.
Survival rates can be measured directly from curve.
Difference between curves can be tested for statistical significance.
Cox Regression Model AKA: Proportional Hazards Survival Model. Used to investigate relationship between an event
(death, recurrence) occurring over time and possible explanatory factors.
Reported result: Hazard ratio (HR). Ratio of the hazard in one group divided the hazard in
another. Interpreted same as risk ratios and odds ratios
HR 1 = no effect HR > 1 increased risk HR < 1 decreased risk
Cox Regression Model Common use in long-term studies
where various factors might predispose to an event. Example: after uterine embolization, which
factors (age, race, uterine size, etc) might make recurrence more likely.
Summary Understanding basic statistical concepts is central to
understanding the medical literature.
Not important to understand the basis of the tests or the underlying math.
Need to know when a test should be used and how to interpret its results