Statistics Flashcards
Type I Error
alpha error = false positive
Statistical test shows that there is a difference when one does not exist
Cause by incorrect stat test or random error
if alpha = 0.05, then 1 in 20 times, a type 1 error will occur even when H0 is rejected
Meaning 5% of the time, a researcher will conclude there is a statistically significant difference when there is not
Type II Error
beta error = false negative
Statistical test concludes there is no difference when one exists
Cause by insufficient power
Large sample size helps decrease chance of error
Typical acceptable error rate = 0.10-0.20
Ordinal data
Qualitative variable (categorical)
Ranked in a specific oder, but no consistent level of magnitude of different between ranks
Example: Likert Scale (strongly agree, agree, neutral, etc); Wong-Baker Faces Pain Rating Scale; Pain rated 0-10); NYHA I, II, III, IV
NOT real numbers
Do not use mean or standard deviation to report
Interval/Ratio Data
Quantitative variables (continuous): can take on any value within a given range
Have a CLEAR numerical value (# hospitalizations, # pregnancies)
Interval has no true 0 but ratio does
Interval: degrees Fahrenheit
Ratio: degrees Kelvin, heart rate, blood pressure, time, distance
Nominal Data
Qualitative variable (categorical)
Data with mutually exclusive categories but no rank or order
Ex: presence of event/disease state (yes/no); gender, race; mortality (dead or alive)
Often expressed as a %
Ex: Pain severity using descriptive terms (minimal, moderate, sharp, aching)
Random Error
Unavoidable, unidentifiable circumstance randomly introduced into a study that is caused by chance or nonsystematic error
Minimize with statistical testing and increased sample size
May impact reliability of results. Can be controlled but not eliminated
Intention to Treat
Once randomized, then analyzed
Maintains integrity of randomization
Conservatively presents results to mimic real world conditions
Preferred type of analysis for superiority trial
Delta Margin
Minimum clinically acceptable difference based on previous research
Used in noninferiority trials
Noninferiority trial
Alternative design when unethical to use placebo
Aim: demonstrate intervention is no worse than control by delta margin
Large sample needed for adequate power
Practice-based Research
Evaluates value of program/service to improve clinical outcomes and/or decrease cost
Confidence Interval
Range of values that probably includes the true treatment effect
Large sample size = narrower, more precise confidence interval
Usually expressed as 95% CI (corresponding to alpha of 0.05)
If continuous variables: 95% CI that includes 0 = not statistically significant
If CI for risk ratio (odds, relative risk, hazards), 95% that includes 1 = non statistically significant
Absolute Risk Reduction/Increase
Difference in risk between control group and intervention group
Relative Risk Reduction/Increase
% reduction in risk in intervention group compared with control group
RRR = (1 - RR) * 100
RRI = (RR - 1) * 100
Relative Risk
Incidence of outcome in exposed group compared with unexposed group
Used in cohort studies
RR < 1: risk of disease lower risk in exposed group
RR = 1: Risk is the same
RR > 1: risk of disease is higher in exposed group
The RISK of someone developing a condition when exposed compared to someone who has NOT been exposed (risk of developing developmental neurologic disorders when exposed to thimerosal compared to someone who was not exposed)
Case Report/Case Series
Observational study - looks at outcome
Case report = 1 patient
Case series = group of patients or a series of case reports
No measure of association
Pro: identifies potential therapies for rare disease, unusual ADRs
Describes innovative approach
Hypothesis generating
Inexpensive, easy to perform
Con: Weakest form of evidence due to lack of study elements that reduce bias. Does not establish causality OR association
CARE guidelines describe what should be in the report
Surrogate Marker
Outcome measure of a lab value, physical biomarker, or other intermediate measure instead of clinical outcome
Convenient
Example: surrogate marker for hypertension is blood pressure
Per protocol (final analysis)
Only patients completing the entire study included in final analysis
Preferred type of analysis for noninferiority
Log Rank Test (Mantel-Cox)
Survival analysis
Compare survival distributions between two or more groups (H0 = no difference in survival between the two populations)
Assesses differences between groups in survival rate
Assumes random sampling, consistent criteria for entry or end point, baseline survival does not change as time progresses, censored subjects that same average survival time as uncensored
Cox Proportional-hazards (cox regression)
Survival analysis
Most popular method to evaluate impact of covariates
Predict time to experience an event taking into account covariates
Allows for calculation of hazard ratio and CI
Kaplan-Meier Survival
Survival analysis
Reflects cumulative proportion of surviving participants and is recalculated every time an event occurs
Estimates proportion of people who would survive a given length of time under the same circumstances
Crossover Clinical Trial
Type of RCTs
Subject serve as own control by receiving all interventions under investigation in a sequential order with washout period between different interventions
Do not use in diseases that are not curable
Do not use if patient cannot return to pretreatment status before each treatment
P value
Probability that results are due to chance, not the intervention
Calculated chance that a type 1 error has occurred
A lower p-value does NOT mean result is more important or meaningful, just that it is statistically significant and not likely to be attributable to chance
Parallel clinical trial
Each subject receives/is assigned to one intervention
Data from all subjects in specific group are pooled together and compared with data from other groups receiving different interventions
+ outcome
intervention – outcome
population
+ outcome
control – outcome
Interventional Study Design
Randomized Controlled Trials
Aim: determine cause and effect by investigating whether differences exist and quantify differences between interventional & control groups
Need to employ methods to minimize risk or error, bias, confounding (ex: blinding, randomization, statistical analysis)
Observational Study Design
Cross-sectional, Case-control, Cohort
Aim: demonstrate association (NOT causation) between exposure and outcome
Can be retrospective or prospective
Prospective cohort > retrospective cohort > case control > cross-sectional
Systematic Error/Bias
Avoidable, identifiable, and non-randomly introduced into a study
Most important way to reduce bias = blinding, randomization
D4 Approach to Biostats
Design of study (independent/parallel or dependent/crossover)
Designated # groups (2 or >2)
Data types (Interval/Ratio, Ordinal, Nominal)
Distribution
Number Needed to Harm
Number of patients needed to treat over a specified period for 1 to experience an adverse event
Number Needed to Treat
Number of patients who would need to be treated over a specified period for 1 patient to be spared a harmful event or experience a beneficial event
NNT = 100/ARR (%) or 1/ARR (decimal)
ARR = Control - intervention (X-Y)
Calculate when there are significant results (!!) for primary outcome (nominal data)
Extrapolation beyond studied time points should NOT occur.
Case-control Study
Observational Study- looks at outcome
Examines individuals with an outcome of interest to determine if there are exposures associated with development of the outcome
Retrospective. The outcome is known at the beginning of the study.
Measure of association: odds ratio
Pro: Good for studying rare outcomes with multiple exposures, esp. unknown risk factors
Pro: Inexpensive, short duration.
Con: Confounding MUST be controlled
Con: Observational and recall bias
Con: Selection bias (see below)
Critical assumptions to minimize bias:
1) cases selected are to be representative of those who have disease. randomly select when possible.
2) controls are representative of general population. identical to cases minus presence of disease.
3) information collected same way for cases & controls
Cross-sectional Study
Observational Study –AKA Prevalence study (snapshot)
Identify the prevalence of a condition in a group of individuals. Studies done by interview, questionnaire, biomedical info.
Measure of association: prevalence
Pro: Provides epidemiology information
Pro: include larger sample size compared with case report
Pro: Include patients regardless of disease severity, access to care
Con: Cannot determine incidence of outcomes or study factors in individual over time
Con: Not ideal for rare exposure, outcomes, or conditions
Ex: prevalence of serious eye disease and visual impairment in north London population
Ex: maternal characteristics and migraine pharmacotherapy during pregnancy
Cohort Study
Observational Study - looks at exposure
Determines ASSOCIATION between exposures/factors and DEVELOPMENT of a disease/condition
Can be prospective or retrospective. In both, need to exclude those with outcome already from the study population.
Measure of Association: relative risk
Retrospective:
-better for rare outcomes (can investigate issues that may have ethical/safety issues in RCT).
-less expensive.
-Con: impacted by confounding variables, recall bias
Prospective:
-can control confounding variables easier.
-can develop temporal relationship.
-Con: more expensive and time consuming.
-Con: more difficult to study rare outcomes than retrospective.
Ex: Framingham Study: prospective cohort of subjects studied over time to evaluate relationship between variety of exposures to development of CV
Ex: Thimerosal DTP: retrospective cohort investigated impact of thimerosal on developmental neurologic disorders
Selection bias
An error in the selection/sampling of individuals for clinical study, which leads to advantage for one group over the other
Impacts case-control studies more than cohort
Performance/interviewer bias
Difference in care provided
Interviews not conducted in a uniform manner
Detection bias
Difference in how the outcome was assessed
Attrition bias
Difference in withdrawal rates from the study
Observational/information bias
Incorrect determination of outcomes or exposures.
Ex: error in recording individual factors for a study (risk factor, timing of blood sample)
Compliance/adherence bias
More subjects in one group fail to follow protocol
Recall bias
Subject in one group more likely to accurately remember facts of interest
“Cases” are more likely to remember exposures than
“controls”
Odds Ratio
Prevalence of EXPOSURE in group with outcome compared with group without outcome
Use in case control study
Interpreted as “odds of exposure to a factor in those with a condition or diseases compared to those who do not have the condition or disease”
OR <1: odds of exposure is lower in diseased group
OR = 1: odds of exposure is same in two groups
OR >1: odds of exposure is greater in the diseased group
If CI crosses 1, then no statistical difference
Intervention = Y
Control = X
Positive outcome
Intervention (Y) = a
Control (X) = c
Negative outcome
(Y) = b
(X) = d
Y = a /(a+b)
X = c/(c+d)
ARR = X - Y
RR = Y/X
RRR = (1 - RR) * 100
OR = (A/C)/(B/D) or (AD)/(BC)
Dependent/Normal/Parametric Stats Test for
Interval/Ratio data
2 groups: Paired t-test
Multiple measures in >=2 groups: repeated measure ANOVA or ANCOVA
Dependent Stats Test for
Ordinal Data
2 groups: Wilcoxon signed rank
Multiple measure in >=2 groups: Friedman
Dependent Stats Test for
Nominal Data
2 groups: McNemar
Multiple measure in >= 2 groups: Cochrane Q
Independent Stats Test for
Interval/Ratio Data
2 groups: t-test
> 2 groups: one -way & two-way ANOVA or ANCOVA
Independent Stats Test for
Ordinal Data
2 groups: Mann-Whitney U (Wilcoxon rank sum)
> 2 groups: Kruskal-Wallis
Independent States Test for
Nominal Data
2 groups: Fischer’s exact
> 2 groups: Chi-square
Mean
A numerical measure of central tendency used in descriptive statistics
Use for continuous & normally distributed data (think interval, ratio)
Arithmetic or geometric
-geometric involves log-normal distributions
Visual methods for descriptive statistics
Frequency distribution
Histogram
Scatterplot
Boxplot
Median
Numerical measure of central tendency used for descriptive statistics
“50th percentile”
Use for ordinal or continuous (interval/ratio) data
Not affected by outliers
Mode
Numerical measure of central tendency used for descriptive statistics
Most common value - sometimes there is more than 1
Can be used for nominal, ordinal, or continuous data
Standard deviation
Numerical measure of variability used for descriptive statistics
Measure of variability about the mean
Only applies to continuous data!! that are normally distributed
Empiric rule for normal distribution:
68% of data found within +/- 1 SD
95% of data found within +/- 2 SD
99% of data found within +/- 3 SD
Coefficient of variation
SD/mean * 100
Relates mean and the standard deviation
Range
Numerical method to describe variability in descriptive statistics
Difference in smallest and largest value in data set (easy calculation) but does not provide much info.
Very sensitive to outliers.
Often reported as actual values (like 0 - 50) instead of range = 50
Percentiles
Numerical measure of variability for descriptive statistics
Ex: 75th percentile; 75% of all values are smaller
Does NOT assume normal distribution
Interquartile range
Numerical measure of variability for descriptive statistics
defined as 25-75th percentile
Frequency distribution
Visual method for descriptive statistics that shows how often a value appears in a set of data
Histogram
Visual method for descriptive statistics that plots distribution of numeric values as a series of bars
Scatterplot
Visual method for descriptive statistics that has dots represent two different numerical values
Box plot
Visual method for descriptive statistics that uses boxes and lines to depict distributions of 1 or more groups of numeric data
Box limits = central 50% of data
Central line = median
Inference
An educated statement about an unknown population
Binomial distribution
Population distribution type.
Discrete distribution.
2 possible outcomes.
Probability of obtaining each outcome is known
You want to know the chance of observing a certain # of successes in a certain # of trials (finite)
Ex: Flipping a coin.
Either heads or tails
Probability of getting tails in 10 tries
Poisson distribution
Population distribution type
Discrete distribution
Counting events in a certain period of observation. Avg # of counts is known
Aim: likelihood of observing a various number of events (infinite)
Probability of ‘r’ events in a population
Ex: How to staff a call center when get x amount of calls in x minutes
Normal distribution
Most common model for population distribution
How to tell if data is normal:
-visually (bell shape)
-Mean & mean will be about equal (nonvisual, but studies may not report both)
-formal test = Kolmogorov-Smirnov
Parametric
Term for normally distributed data
Parameters, mean, and SD completely define distribution of data
Probability
Likelihood that any one event will occur given all the possible outcomes
Distribution of means
If you pull separate samples from a single population in normally distributed data, the means will be slightly different
However if you take the mean of the ‘distribution of the means’, it should be equal to unknown population mean
Central limit theorem
The distribution of means from random samples is about normal regardless of underlying population distribution
Standard Error of the Mean (SEM)
Standard deviation of means in distribution of means
SEM = standard deviation / square root of n (sample size)
Quantifies uncertainty in the estimate of the mean, which is important for hypothesis testing and 95% CI estimation
95% vs 90% confidence interval
95% will always be wider, so it is more likely to encompass the true population mean
95% CI = mean +/- 1.96 * SEM
90% CI = mean +/- 1.64 * SEM
Null hypothesis
H0 = states no difference between groups being compared
Results of hypothesis testing:
Reject H0 = statistically significant difference between groups (unlikely attributable to chance)
Accept H0 = no statistically significant difference between groups
Alternative hypothesis
HA = states that there is a difference between groups being compared
Nondirectional, difference hypothesis test
Asks ‘‘are the means different?’’
Use traditional 2 sided t test & CI
Nondirectional, equivalence hypothesis test
Asks ‘‘are the means practically equivalent?’’
Use two 1-sided t-test (TOST) & CI
Directional, superiority hypothesis test
Asks ‘‘is mean 1 > mean 2?”
Use traditional 1-sided t-test & CI
Directional, noninferiority hypothesis test
Asks ‘‘is mean 1 no more than a certain amount lower than mean 2?’’
Use CI
Power
power = 1-B (probability of making a type II error)
Dependent on:
-predetermined alpha
-sample size
-desired effect size
-variability of outcomes you want to measure
Decreased by poor study design, small sample size, incorrect statistical tests
Effect size
Size of difference between outcomes
Necessary components to estimate sample size
-Acceptable type II error rate (0.10-0.20)
-Observed difference in predicted study outcomes that is clinically significant AND its expected variability
-Acceptable type I error rate (0.05)
-Statistical test used for primary end point
Parametric test
Assumes:
-Normal or near normal underlying distribution (mean ~ median)
-QUANTITATIVE CONTINUOUS DATA (INTERVAL OR RATIO)
-Investigated data have homoscedasticity
Homoscedasticity
Data being investigated have variances that are homogenous between groups
Important for parametric tests
Nonparametric tests
Data are NOT normally distributed
May be skewed quantitative continuous data, quantitative (discrete) data, or qualitative (ordinal/nominal) data
Correlation
Examines strength and direction of association between two variables
Correlation does not reflect one variable is useful in predicting the other (correlation does not equal causation)
Closer ‘r’ is to 1, the more highly correlated the variables are
Closer ‘r’ is to 0, the weaker the relationship
AKA degree of association
Visual inspection of scatterplot is ESSENTIAL before using correlation analysis
Regression
Examines ability of one or multiple variables to predict a dependent variable
Commonly used to determine whether differences exist btwn groups after controlling for confounding variables
Purposes: develop prediction model & estimate accuracy of prediction
Pearson correlation
Correlation test that is a measure of strength of relationship between two CONTINUOUS variables that are normally distributed & linearly related
Hypothesis test determines whether correlation coefficient is different from 0 – highly influenced by sample size
Spearman rank correlation
Nonparametric correlation test of the strength of a monotonic association (linear or nonlinear) between two CONTINUOUS variables.
Can be used for ordinal data as well.
Point-biserial correlation
Nonparametric correlation test of strength and direction of association between one dicotomous variable (nominal) and one continuous variable (I/R)
Coefficient of determination
r2 or R2 = used to describe how well regression analysis was predicted (extent of variability in dependent variable that can be explained by independent variable)
Range from 0 to1
r2 of 0.80 = 80% of variability in Y is explained by variability in X
Does not provide info for relationship of X and Y, rather describes how clearly a regression model worked.
Multiple linear regression analysis
Regression analysis
One continuous dependent variable + two or more continuous or categorical independent variables
Aim: find effect of one or more variables on a dependent variable while controlling for the effects of other independent variables
ANCOVA
Multiple regression model
Continuous & categorical independent variables
Aim: determine effect of one or more categorical variables (factors) on a dependent variable while controlling for effects of one or more continuous variables (covariates)
Simple logistic regression
Regression model
One categorical dependent response variable and one continuous or categorical explanatory variable
Multiple logistic regression
Regression model
Oen categorical dependent response variable and two or more continuous or categorical explanatory variables
Aim: discern the effect of one or more variables on a dependent variable while controlling for effect of covariates
Nonlinear regression
Regression model
Variables are not linearly related
PK equations derived from here
Polynomial regression
Regression model
Any number of response and continuous variables with a curvilinear relationship (cubed, squared)
y=mx + b
linear regression
Y = dependent variable
m = slope
x = independent variable
b = y intercept
Survival analysis
Studies the time between entry in a study and some event (death, MI)
Quasi-experimental study
Evaluate interventions and causality but are NOT randomized
Internal validity
Degree to which the outcome can be explained by differences in the assigned groups
Related to study methods (proper design, conduction, analysis)
Factors that affect internal validity:
-poor study design
-inadequate randomization
-lack of/inappropriate blinding
-Using inaccurate measurements
-Using inappropriate statistical methods
-Incomplete outcome reporting
Occurs more in nonrandomized or observational studies
External validity
The degree to which findings can be generalized to a population beyond the study
Factors that affect external validity (6 S’s):
-Setting
-Selection of patients (inclusion/exclusion, placebo/treatment)
-Study patient characteristics (clinical characteristics, race/sex, uniformity of pathology, comorbidities, severity of disease)
-Selected trial protocol is not same as routine practice (intervention timing, appropriateness of control, frequency of monitoring)
-Study outcome measures & follow up (accepting surrogate markers, reproducibiilty of findings, frequency/adequacy of follow up
-Side effects (discontinuation rates, completeness of ADR reporting, intensity of safety procedures
Misclassification bias
Subject is categorized into incorrect group
Differential bias
Type of misclassification bias when information errors differ between groups
Ex: in cohort, difference between those with disease and those without
Nonrandom error
Non-differential bias
Type of misclassification bias when results collected are incorrect, but affect both groups the same
Systematic error
Confounding variables
Nonrandomized variable
Affects independent or dependent variable - unable to determine true effect on measured outcome (may hide OR exaggerate true association)
Minimize:
-randomize
-match subjects in analysis by stratification, propensity score matching, or multivariable analysis techniques
Point prevalence vs period prevalance
Point prevalence: prevalence on a given date
Period prevalence: prevalence in a period (year, month)
Hazards ratio
Estimates risk at any given point in time within a certain time period
HR <1: lower risk of the event in experimental group than in control (experimental treatment better than control treatment)
HR = 1: event rates are the same in both groups
HR >1: greater risk of event in experimental group than in control group (experimental treatment is worse than control treatment)
RR/OR = 0.75?
RR/OR = 3.0?
0.75:
RR = 25% reduction in risk (1-0.75)
OR = odds are 0.75/1
3.0:
RR = 200% (3x increase in risk)
OR = odds are 3/1 higher
So for RR, take the RR value - 1 to get the reduction or increase in risk
OR will be the OR value/1
Types of blinding (single, double, triple, double dummy, open)
Single: either subject or investigator blinded
Double: both subject and investigator blinded
Triple: subject, investigator, and analysis group blinded
Double dummy: match active & control groups when difference in delivery (IV or PO, will get placebo of the opposite)
Open label: everyone aware
Types of randomization (block, stratification, cluster)
Block: divide groups into blocks, then randomize
Stratification: group based on similar characteristics
Cluster: randomly assign groups, not individuals
Types of treatment controls (active, historical)
Active: compare experimental with established treatment
Historical: compare new treatment to a group of patients treated in the past
Factorial design
Answers two separate research questions in a single group of subjects
Used in RCTs
Composite end point
Used in RCTs - combines several events into 1 event category (CV events) - even combining morbidity and mortality
Results for each individual end point within composite should also be reported
Several limitations
-difficult to interpret
-dilute effect
-average overall effect (if component end points move in opposite directions, the overall composite would be averaged)
Benefits
-increase # of events, so can reduce sample size & cost (good for investigator)
-do not need multiple tests
As-treated analysis
Type of analysis where subjects are analyzed by actual intervention received
If assigned to active treatment, but did not take active treatment, then analyzed as if in placebo group
Destroys randomization process. Use with caution.
Narrative review
Summarizes several studies, but no systematic methods.
Subjective.
Ex: standard literature review
Qualitative Systematic Review
Comprehensive literature search using explicit methods (inclusion/exclusion criteria), critically appraise it, and synthesize
Objective
Includes systematic review.
Quantitative Systematic Review
AKA Meta analysis
Systematic review using statistical techniques to summarize the results of all studies evaluated
Details of each study are essential – relies on criteria for inclusion of previous studies & statistical methods to ensure validity
Use FOREST PLOT to summarize
Heterogeneity
use in Meta Analysis
states the degree of variation or difference in results across several studies included in analysis. Do not want a lot of heterogeneity.
Common tests: Chi2, Cochran Q, I2
i2 < 25% = low heterogeneity = studies similar
i2 25-50% = moderate heterogeneity = caution needed
i2 50% = substantial heterogeneity = difficult to draw conclusions from meta analysis
Chi2 = p value for null hypothesis that there is no heterogeneity in the studies
If p <0.01, then reject null and assume there is heterogeneity
funnel plot
assesses publication bias in meta analysis studies
Y axis = study precision (standard error)
X axis = estimate of effect of each study
Graph should look like an inverted funnel, because higher sample size studies will have higher effect size
Asymmetrical plot = publication bias
**look at examples
Absolute risk
Chance of an outcome occurring
Absolute risks are more important than relative risks
Absolute risk difference (reduction/increase)
Difference in the absolute risk (chance of outcome occurring) in exposed vs unexposed group
Guidelines for clinical trials
CONSORT = clinical trials
STROBE = observational studies
PRISMA = meta-analysis and systematic reviews
EQUATOR = international guidelines
Cost minimization
Pharmacoeconomic study that shows the difference in cost among comparable therapies are evaluated
therapies must have similar outcomes
Cost effectiveness
Pharmacoeconomic study to measure the cost impact when health outcomes are improved
ex: years of life saved, number of symptom free days, blood glucose, blood pressure
Cost utility
Pharmacoeconomic study that compares outcomes related to mortality when mortality may not be the most important outcome
Ex: quality adjusted life years (QALY)
Cost benefit
Pharmacoeconomic study that analyzes cost of treatment and cost saved with beneficial outcomes
Sensitivity vs specificity
Sensitivity: proportion of TRUE POSITIVES that are CORRECTLY identified by test. high sensitivity = negative test can rule out disorder
Calculated as true positive/(true positive + false positive)
Specificity: proportion of TRUE NEGATIVES that are CORRECTLY identified in a test. High specificity = positive test can rule IN the disorder
Calculated as true negative/(true negative + false negative)
