introduction to biostatistics for clinical and translational researchers kumc departments of...
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Introduction to Biostatistics for Clinical and Translational
Researchers
KUMC Departments of Biostatistics & Internal Medicine
University of Kansas Cancer Center
FRONTIERS: The Heartland Institute of Clinical and Translational Research
Course Information
Jo A. Wick, PhDOffice Location: 5028 RobinsonEmail: [email protected]
Lectures are recorded and posted at http://biostatistics.kumc.edu under ‘Events & Lectures’
Objectives
Understand the role of statistics in the scientific process and how it is a core component of evidence-based medicine
Understand features, strengths and limitations of descriptive, observational and experimental studies
Distinguish between association and causationUnderstand roles of chance, bias and
confounding in the evaluation of research
Course Calendar
July 5: Introduction to Statistics: Core ConceptsJuly 12: Quality of Evidence: Considerations for
Design of Experiments and Evaluation of LiteratureJuly 19: Hypothesis Testing & Application of
Concepts to Common Clinical Research QuestionsJuly 26: (Cont.) Hypothesis Testing & Application
of Concepts to Common Clinical Research Questions
“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”
Albert Einstein (1879-1955)
Vocabulary
Basic Concepts
Statistics is a collection of procedures and principles for gathering data and analyzing information to help people make decisions when faced with uncertainty.
In research, we observe something about the real world. Then we must infer details about the phenomenon that produced what we observed.
A fundamental problem is that, very often, more than one phenomenon can give rise to the observations at hand!
Example: Infertility
Suppose you are concerned about the difficulties some couples have in conceiving a child.It is thought that women exposed to a particular
toxin in their workplace have greater difficulty becoming pregnant compared to women who are not exposed to the toxin.
You conduct a study of such women, recording the time it takes to conceive.
Example: Infertility
Of course, there is natural variability in time-to-pregnancy attributable to many causes aside from the toxin.
Nevertheless, suppose you finally determine that those females with the greatest exposure to the toxin had the most difficulty getting pregnant.
Example: Infertility
But what if there is a variable you did not consider that could be the cause?
No study can consider every possibility.
Example: Infertility
It turns out that women who smoke while they are pregnant reduce the chance their daughters will be able to conceive because the toxins involved in smoking effect the eggs in the female fetus.
If you didn’t record whether or not the females had mothers who smoked when they were pregnant, you may draw the wrong conclusion about the industrial toxin.
Fertility
Natural Variability
Smoking Behaviors of Mother
Environmental Toxins
Example: Infertility
Exposed to Toxin
Majority exposed to
smoke in womb
Prolonged time-to-
conceive found
Unexposed to Toxin
Majority unexposed to
smoke in womb
Time-to-conceive measured
??
Type I Error!
Lurking (Confounding) Variable → Bias
Example: Infertility
Exposed to Toxin
Some smoking exposure
An insignificant change in time-to-
conceive found
Unexposed to Toxin
Some smoking exposure
Time-to-conceive measured
??
Type II Error!
Lurking (Confounding) Variable → “Noise”
The Role of Statistics
The conclusions (inferences) we draw always come with some amount of uncertainty due to these unobserved/unanticipated issues.
We must quantify that uncertainty in order to know how “good” our conclusions are.
This is the role that statistics plays in the scientific process.P-values (significance levels)Level of confidenceStandard errors of estimatesConfidence intervalsProper interpretation (association versus causation)
The Role of Statistics
Scientists use statistical inference to help model the uncertainty inherent in their investigations.
xxx
x
1
2
3
n
population XS
goal: statistical inference
(uncertainty measured by probability)
histogram(observation)
sample
(reality)
?
(imagination)popula tion model
Evidence-based Medicine
Evidence-based practice in medicine involvesgathering evidence in the form of scientific data.applying the scientific method to inform clinical
practice, establishment or development of new therapies, devices, programs or policies aimed at improving health.
Types of Evidence
Scientific evidence: “empirical evidence, gathered in accordance to the scientific method, which serves to support or counter a scientific theory or hypothesis”Type I: descriptive, epidemiologicalType II: intervention-based Type III: intervention- and context-based
Evidence-based Medicine
Evidence-based practice results in a high likelihood of successful patient outcomes and more efficient use of health care resources.
The Scientific Method
Revise
Experiment
Observe
Clinical Evaluation
Revise Design & Hypothe
sis
Run Experimen
t
Evidence (Data)
Types of Studies
Purpose of research1) To explore
2) To describe or classify
3) To establish relationships
4) To establish causalityStrategies for accomplishing these purposes:
1) Naturalistic observation
2) Case study
3) Survey
4) Quasi-experiment
5) Experiment
Am
big
uit
y Co
ntro
l
Generating Evidence
Studies
Descriptive Studies
Populations Individuals
Case Reports
Case Series
Cross Sectional
Analytic Studies
Observational
Case Control Cohort
Experimental
RCT
Complexity and Confidence
Observation versus Experiment
A designed experiment involves the investigator assigning (preferably randomly) some or all conditions to subjects.
An observational study includes conditions that are observed, not assigned.
Example: Heart Study
Question: How does serum total cholesterol vary by age, gender, education, and use of blood pressure medication? Does smoking affect any of the associations?
Recruit n = 3000 subjects over two yearsTake blood samples and have subjects answer a
CVD risk factor surveyOutcome: Serum total cholesterolFactors: BP meds (observed, not assigned)Confounders?
Example: Diabetes
Question: Will a new treatment help overweight people with diabetes lose weight?
N = 40 obese adults with Type II (non-insulin dependent) diabetes (20 female/20 male)
Randomized, double-blind, placebo-controlled study of treatment versus placebo
Outcome: Weight lossFactor: Treatment versus placebo
How to Talk to a Statistician?
“It’s all Greek to me . . .”
Καλημέρα
Why Do I Need a Statistician?
Planning a studyProposal writingData analysis and interpretationPresentation and manuscript development
When Should I Seek a Statistician’s Help?
Literature interpretationDefining the research questionsDeciding on data collection instrumentsDetermining appropriate study size
What Does the Statistician Need to Know?
General idea of the researchSpecific Aims and hypotheses would be ideal
What has been done beforeLiterature review!Outcomes under considerationStudy populationDrug/Intervention/Device
Rationale for the studyBudget constraints
“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”
Albert Einstein (1879-1955)
Vocabulary
Hypotheses: a statement of the research question that sets forth the appropriate statistical evaluationNull hypothesis “H0”: statement of no differences or
association between variablesAlternative hypothesis “H1”: statement of differences
or association between variables
Disproving the Null
If someone claims that all swans are white, confirmatory evidence (in the form of lots of white swans) cannot prove the assertion to be true.
Contradictory evidence (in the form of a single black swan) makes it clear the claim is invalid.
The Scientific Method
Observation
Hypothesis
Experiment
Results
Evidence supports H
Evidence inconsistent
with H
Revise H
Hypothesis Testing
By hypothesizing that the mean response of a population is 26.3, I am saying that I expect the mean of a sample drawn from that population to be ‘close to’ 26.3:
x
Px
24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0
Hypothesis Testing
What if, in collecting data to test my hypothesis, I observe a sample mean of 26?
What conclusion might I draw?
x
Px
24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0
x
Px
24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0
Hypothesis Testing
What if, in collecting data to test my hypothesis, I observe a sample mean of 27.5?
What conclusion might I draw?
x
Px
24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0
Hypothesis Testing
What if, in collecting data to test my hypothesis, I observe a sample mean of 30?
What conclusion might I draw?
?
Hypothesis Testing
If the observed sample mean seems odd or unlikely under the assumption that H0 is true, then we reject H0 in favor of H1.
We typically use the p-value as a measure of the strength of evidence against H0.
What is a P-value?
x
Px
24.5 25.0 25.5 26.0 26.5 27.0 27.5 28.0
Null distribution
Observed sample mean
p-value
A p-value is the area under the curve for values of the sample mean more extreme than what we observed in the sample we actually gathered.
If H1 states that the mean is greater than 26.3, the p-value is as shown.If H1 states that the mean is different
than 26.3, the p-value is twice the area shown, accounting for the area in both tails.
If H1 states that the mean is less than 26.3, the p-value is the area to the left of the observed sample mean.
A p-value the probability of getting a sample mean as favorable or more favorable to H1 than what was observed, assuming H0 is true.
The tail of the distribution it is in is determined by H1.
Vocabulary
One-tailed hypothesis: outcome is expected in a single direction (e.g., administration of experimental drug will result in a decrease in systolic BP)
Two-tailed hypothesis: the direction of the effect is unknown (e.g., experimental therapy will result in a different response rate than that of current standard of care)
Vocabulary
Type I Error (α): a true H0 is incorrectly rejected“An innocent man is proven GUILTY in a court of law”Commonly accepted rate is α = 0.05
Type II Error (β): failing to reject a false H0
“A guilty man is proven NOT GUILTY in a court of law”Commonly accepted rate is β = 0.2
Power (1 – β): correctly rejecting a false H0
“Justice has been served”Commonly accepted rate is 1 – β = 0.8
Decisions
ConclusionTruth
H1 H0
H1 Correct: Power Type I Error
H0 Type II Error Correct
Statistical Power
Primary factors that influence the power of your study:Effect size: as the magnitude of the difference you wish
to find increases, the power of your study will increaseVariability of the outcome measure: as the variability
of your outcome decreases, the power of your study will increase
Sample size: as the size of your sample increases, the power of your study will increase
Statistical Power
Secondary factors that influence the power of your study:DropoutsNuisance variationConfounding variablesMultiple hypothesesPost-hoc hypotheses
Hypothesis Testing
We will cover these concepts more fully when we discuss Hypothesis Testing and Quality of Evidence
Descriptive Statistics
Field of Statistics
Statistics
Descriptive Statistics
Methods for processing,
summarizing, presenting and describing data
Experimental Design
Techniques for planning and conducting
experiments
Inferential Statistics
Evaluation of the information
generated by an experiment or
through observation
Field of Statistics
Statistics
Descriptive
Graphical Numerical
Inferential
Estimation Hypothesis Testing
Experimental Design
Field of Statistics
Descriptive statisticsSummarizing and describing the dataUses numerical and graphical summaries to characterize
sample dataInferential statistics
Uses sample data to make conclusions about a broader range of individuals—a population—than just those who are observed (a sample)
The principal way to guarantee that the sample
population sample
Field of Statistics
Experimental DesignFormulation of hypothesesDetermination of experimental conditions,
measurements, and any extraneous conditions to be controlled
Specification of the number of subjects required and the population from which they will be sampled
Specification of the procedure for assigning subjects to experimental conditions
Determination of the statistical analysis that will be performed
Descriptive Statistics
Descriptive statistics is one branch of the field of Statistics in which we use numerical and graphical summaries to describe a data set or distribution of observations.
Statistics
Descriptive
Graphs Statistics
Inferential
Hypothesis Testing
Interval Estimates
Types of Data
All data contains information.It is important to recognize that the hierarchy
implied in the level of measurement of a variable has an impact on (1) how we describe the variable data and
(2) what statistical methods we use to analyze it.
Levels of Measurement
Nominal: differenceOrdinal: difference, orderInterval: difference, order, equivalence of intervalsRatio: difference, order, equivalence of intervals,
absolute zero
discrete qualitative
continuous quantitative
Types of Data
NOMINAL
ORDINAL
INTERVAL
RATIO
Information increases
Ratio Data
Ratio measurements provide the most information about an outcome.
Different values imply difference in outcomes.6 is different from 7.
Order is implied.6 is smaller than 7.
Ratio Data
Intervals are equivalent.The difference between 6 and 7 is the same as the
difference between 101 and 102.Zero indicates a lack of what is being measured.
If item A weighs 0 ounces, it weighs nothing.
Ratio Data
Ratio measurements provide the most information about an outcome.
Can make statements like: “Person A (t = 10 minutes) took twice as long to complete a task as Person B (t = 5 minutes).”
This is the only type of measurement where statements of this nature can be made.
Examples: age, birth weight, follow-up time, time to complete a task, dose
Interval Data
Interval measurements are one step down on the “information” scale from ratio measurements.Difference and order are implied and intervals
are equivalent.BUT, zero no longer implies an absence of the
outcome.What is the interpretation of 0C? 0K?The Celsius and Fahrenheit scales of temperature are
interval measurements, Kelvin is a ratio measurement.
Interval Data
Interval measurements are one step down on the “information” scale from ratio measurements.You can tell what is better, and by how much, but
ratios don’t make sense due to the lack of a ‘starting point’ on the scale.60F is greater than 30F, but not twice as hot since 0F
doesn’t represent an absence of heat.Examples: temperature, dates
Ordinal Data
Ordinal measurements are one step down on the “information” scale from interval measurements.
Difference and order are implied.BUT, intervals are no longer equivalent.
For instance, the differences in performance between the 1st and 2nd ranked teams in basketball isn’t necessary equivalent to the differences between the 2nd and 3rd ranked teams.
The ranking only implies that 1st is better than 2nd, 2nd is better than 3rd, and so on . . . but it doesn’t try to quantify the ‘betterness’ itself.
Ordinal Data
Ordinal measurements are one step down on the “information” scale from interval measurements.Examples: Highest level of education achieved,
tumor grading, survey questions (e.g., likert-scale quality of life)
Nominal Data
Nominal measurements collect the least amount of information about the outcome.Only difference is implied.Observations are classified into mutually
exclusive categories.Examples: Gender, ID numbers, pass/fail
response
Levels of Measurement
It is important to recognize that the hierarchy implied in the level of measurement of a variable has an impact on (1) how we describe the variable data and
(2) what statistical methods we use to analyze it. The levels are in increasing order of mathematical
structure—meaning that more mathematical operations and relations are defined—and the higher levels are required in order to define some statistics.
Levels of Measurement
At the lower levels, assumptions tend to be less restrictive and the appropriate data analysis techniques tend to be less sensitive.
In general, it is desirable to have a higher level of measurement.
A summary of the appropriate statistical summaries and mathematical relations or operations is given in the next table.
Levels of Measurement
Level Statistical SummaryMathematical
Relation/Operation
Nominal Mode one-to-one transformations
Ordinal Median monotonic transformations
Interval Mean, Standard Deviation positive linear transformations
Ratio Geometric Mean, Coefficient of Variation multiplication by c 0
We must know where an outcome falls on the measurement scale--this not only determines how we describe the data (descriptive statistics) but how we analyze it (inferential statistics).
Using Graphs to Describe Data
Nominal and ordinal measurements are discrete and qualitative, even if they are represented numerically.Rank: 1, 2, 3Gender: male = 1, female = 0
We typically use frequencies, percentages, and proportions to describe how the data is distributed among the levels of a qualitative variable.
Bar and pie charts are even more useful.
Example: Myopia
A survey of n = 479 children found that those who had slept with a nightlight or in a fully lit room before the age of 2 had a higher incidence of nearsightedness later in childhood.
No Myopia
Myopia High Myopia
Total
Darkness 155 (90%) 15 (9%) 2 (1%) 172 (100%)
Nightlight 153 (66%) 72 (31%) 7 (3%) 232 (100%)
Full Light 34 (45%) 26 (48%) 5 (7%) 75 (100%)Total 342 (71%) 123 (26%) 14 (3%) 479
(100%)
Example: Myopia
Darkness
Nightlight
Full Light
0 10 20 30 40 50 60 70 80 90 100
High
Some
None
Example: Myopia
As the amount of sleep time light increases, the incidence of myopia increases.
This study does not prove that sleeping with the light causes myopia in more children.
There may be some confounding factor that isn’t measured or considered-possibly genetics.Children whose parents have myopia are more likely to
suffer from it themselves.It’s also possible that those parents are more likely to
provide light while their children are sleeping.
Example: Nausea
How many subjects experienced drug-related nausea?
Nausea No Nausea0
2
4
6
8
10
12
0 mg 10 mg 20 mg 50 mg
Dose Nausea No Nausea 0 mg 0 9 10 mg 1 10 20 mg 3 10 50 mg 3 11
Example: Nausea
With unequal sample sizes across doses, it is more meaningful to use percent rather than frequency.
Nausea No Nausea0%
10%20%30%40%50%60%70%80%90%
100%
0 mg 10 mg 20 mg 50 mg
Dose Nausea No Nausea 0 mg 0 (0%) 9 (100%) 10 mg 1 (9%) 10 (91%) 20 mg 3 (23%) 10 (77%) 50 mg 3 (21%) 11 (79%)
Bar & Pie Charts
Race Percent
Caucasian 30
African American 20
Hispanic 17
Asian American 13
Native American 13
Other 7
Caucasian
African AmericanHispanic
Asian American
Native American
Other
Ethnicity
Caucasian African American Hispanic Asian American Native American Other
05
10
15
20
25
30
Using Graphs to Describe Data
Interval and Ratio variables are continuous and quantitative and can be graphically and numerically represented with more sophisticated mathematical techniques.HeightSurvival Time
We typically use means, standard deviations, medians, and ranges to describe how the variables tend to behave.
Histograms and boxplots are even more useful.
Example: Time-to-death
Suppose that we record the variable x = time-to-death of n = 100 patients in a study.
Time
x
Fre
quen
cy
0 5 10 15
010
20
30
40
Example: Time-to-death
We can quickly observe several characteristics of the data from the histogram:For most subjects, death occurred between 0 and 5
monthsFor a few subjects, death occurred past 15 months
From this picture, we may wish to identify the distinguishing characteristics of the individuals with unusually long times.
Example: Weight
Suppose we record the weight in pounds of n = 100 subjects in a study.
1 1.5Q IQR 3 1.5Q IQR
IQR
1Q 3Q2Q
**
outlier outlier
x
Example: Tooth Growth
Boxplots represent the same information, but are more useful for comparing characteristics between several data sets.
Right: distributions of tooth growth for two supplements and three dose levels
Using Numbers to Describe Data
Nominal and ordinal measurements are discrete and qualitative, even if they are represented numerically.Rank: 1, 2, 3Gender: male = 1, female = 0
Interval and Ratio variables are continuous and quantitative and can be graphically and numerically represented with more sophisticated mathematical techniques.HeightSurvival Time
Using Numbers to Describe Data
Nominal and ordinal measurements are qualitative, even if they are represented numerically. We typically describe qualitative data using frequencies
and percentages in tables.Measures of central tendency and variability don’t
make as much sense with categorical data, though the mode can be reported.
Describing Data
Interval and ratio measurements are quantitative. When dealing with a quantitative measurements, we typically describe three aspects of its distribution.Central tendency: a single value around which data
tends to fall.Variability: a value that represents how scattered the
data is around that central value--large values are indicative of high scatter.
We also want to describe the shape of the distribution of the sample data values.
Central Tendency
location
Mean: arithmetic average of dataMedian: approximate middle of dataMode: most frequently occurring value
Central Tendency
Mode, MoThe most frequently occurring value in the data set.May not exist or may not be uniquely defined.It is the only measure of central tendency that can be
used with nominal variables, but it is also meaningful for quantitative variables that are inherently discrete (e.g., performance of a task).
Its sampling stability is very low (i.e., it varies greatly from sample to sample).
Central Tendency: ModeHistogram of x
x
Den
sity
0 5 10 15
0.0
00.0
50.1
00.1
50.2
0
Mo
Central Tendency: Mode
Males
Females
0 2 4 6 8 10 12 14 16
Mo
Central Tendency
Median, MThe middle value (Q2, the 50th percentile) of the variable.It is appropriate for ordinal measures and for skewed
interval or ratio measures because it isn’t affected by extreme values.
It’s unaffected (robust to outliers) because it takes into account only the relative ordering and number of observations, not the magnitude of the observations themselves.
It has low sampling stability.
Example: Median
Suppose we have a set of observations:
1 2 2 4The median for this set is M = 2.
Now suppose we accidentally mismeasured the last observation:
1 2 2 9The median for this new set is still M = 2.
Central Tendency: MedianHistogram of x
x
Den
sity
0 5 10 15
0.0
00.0
50.1
00.1
50.2
0
Mo M
Central Tendency
Mean, The arithmetic average of the variable x.It is the preferred measure for interval or ratio variables
with relatively symmetric observations.It has good sampling stability (e.g., it varies the least
from sample to sample), implying that it is better suited for making inferences about population parameters.
It is affected by extreme values because it takes into account the magnitude of every observation.
It can be thought of as the center of gravity of the variable’s distribution.
x
Example: Mean
Suppose we have a set of observations:
1 2 2 4The median for this set is M = 2, the mean is
Now suppose we accidentally mismeasured the last observation:
1 2 2 9The median for this new set is still M = 2, but the
new mean is
2.25.x
3.5.x
Central Tendency: MedianHistogram of x
x
Den
sity
0 5 10 15
0.0
00.0
50.1
00.1
50.2
0
Mo M x
Variability
spread
Range: difference between min and max values
Standard deviation: measures the spread of data about the mean, measured in the same units as the data
Variability
Measures of variability depict how similar observations of a variable tend to be.
Variability of a nominal or ordinal variable is rarely summarized numerically.
The more familiar measures of variability are mathematical, requiring measurement to be of the interval or ratio scale.
Variability
Range, RThe distance from the minimum to the maximum
observation.Easy to calculate.Influenced by extreme values (outliers).
1 2 3 4 10 R = 10 - 1 = 9
1 2 3 4 100 R = 100 - 1 = 99
Variability
Interquartile Range, IQRThe distance from the 1st quartile (25th percentile) to the
3rd quartile (75th percentile), Q3 - Q1.Unlike the range, IQR is not influenced by extreme
values.
Variability: IQR
1 1.5Q IQR 3 1.5Q IQR
IQR
1Q 3Q2Q
**
outlier outlier
x
Variability
Standard deviation, sRepresents the average spread of the data around the
mean.Expressed in the same units as the data.“Average deviation” from the mean.
Variability
Variance, s2
The standard deviation squared.“Average squared deviation” from the mean.
Shape
shape
Distribution Shapes
Summary
Basic ConceptsDefinition and role of statisticsVocabulary lesson
• Brief introduction to Hypothesis Testing• Brief introduction to Design concepts
Descriptive StatisticsLevels of MeasurementGraphical summariesNumerical summaries
Next time: Study Design Considerations and Quality of Evidence