Stats Flashcards
Why do we need statistics?
Statistics allows us to draw from the sample, conclusions about the general population.
The Central Limit Theorem
The sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.
The Gaussian Distribution
1SD - 68%
2SD - 95%
3SD - 99.7%
Parametric Statistics
A class of statistical procedures that rely on assumptions about the shape of the distribution (assume normal) in the underlying population and about the form or parameters (means, SD) of the assumed distribution.
Non-Parametric
A class of statistical procedures that do not rely on assumptions about the shape or form of the probability from which the data were drawn.
Measures of Location
Mean
Median
Mode
Measures of Dispersion
Range Variance Standard Deviation Standard Error Confidence Interval
Left Skewed (Negative)
Mean –> Median –> Mode
Right Skewed (Positive)
Mode –> Median –> Mean
Boxplots
Largest observed Value that is not an outlier
75th percentile
Median
25th percentile
Smallest observed value that is not an outlier
Range
The difference between the largest and smallest sample values. (Without outliers)
Variance
The average of the square distance of each value from the mean
S^2 = [Sigma(X-M)^2] / [N-1]
Reliable but not user friendly so not often reported.
Standard Deviation
Tells you how tightly each sample is clustered around the mean.
Square root of the variance
Standard Error
Measure of how far the sample mean is away from the population mean.
SEM = SD/sqrt(N)
When to use SD vs SEM?
SD - If the scatter is caused by biological variability and you want to show that variability.
SEM - If the variability is caused by experimental imprecision and you want to show the precision of the calculated mean.
Confidence Interval
An estimate of the range that is likely to contain the true population mean.
CI = X +/- (SEM x Z)
X: sample mean ; Z: 1.96, critical value for normal distribution
If it includes zero, invalid.
Outliers
Grubb’s Test
Z = (mean - value)/SD
If greater than Z tab, it is an outlier. If less than Z tab, must keep it.
Precision
The degree to which repeated measurements under unchanged conditions show the same results. High precision lowers SD.
Accuracy
The degree of closeness of measurements of a quantity to that quantity’s true value. High accuracy reflects the true population mean.
Repeatability
Same as precision (if you can repeat it)
Reproducibility
The ability of an entire experiment or study to be duplicated, either by the same researcher or by someone else working independently.
Sources of Variability
Random error
Systematic error
Null Hypothesis (Ho)
States that there is NO difference between groups. A Study is designed to disprove this assertion by testing for a statistically significant difference b/w A and B (this is called the alternate hypothesis)
Presumed true until statistical evidence proves otherwise.
Ho: u1 = u2
Alternative Hypothesis (Ha)
There IS a treatment difference between groups. If you fail to accept (or reject) Ho, you are accepting the alternative hypothesis.
Type I Error
alpha
Occurs when Ho is true, but it is rejected in error.
False positive.
When a p-value is
Type II Error
beta
Occurs when Ho is false, yet it is accepted in error.
False negative.
Power = 1 - beta
z - statistic
A z-test is any statistical test for which the distribution of the test statistic can be approximated by a normal distribution. Because of the central limit theorem, many test stats are approximately normally distributed for large samples (n>30).
z = (x-u)/ [SD/sqrt(n)]
Then you compare it to a z table of critical values and find the probability of getting greater than or equal to a z value.
t - statistic
Similar to z-test.
Use when n
p - value
The probability that the result obtained was due to chance.
Power
The probability that the test will reject the null hypothesis when the null hypothesis is false (avoiding type II error).
Power = 1 - beta
A higher statistical power means that we can be more certain that the null hypothesis was correctly rejected.
Student t - test
N
N ttab , then reject Ho and conclude that the sample means are significantly different.
Degrees of Freedom
If you have an N of 4, you have 3 degrees of freedom
(N-1)
df = 2N-2 ?
Paired t-test
The observed data are from the same subject or from a matched subject and are drawn from a population with a normal distribution.
Unpaired t-test
The observed data are from two independent, random samples from a population with a normal distribution.
One tailed t-test
Will test either if the mean is significantly greater than x or if less than x, but not both. Provides more power to detect an effect in one direction by not testing the effect in the other direction.
Two tailed t-test
More robust than 1 tailed
Will test both if the mean is significantly greater than or less than x.
ANOVA
ANalysis Of VAriance
- To compare three or more means
- We use sum of squares
F = MS(bg) / MS (wg) –> F-statistic
signal:noise ratio
The higher the F value, the more likely you can reject Ho that the means are equal. Table will give you corresponding p value?
One way ANOVA
1 measurement variable and 1 nominal variable
Ex: measure glycogen content for multiple samples of heart, liver, kidney, lung, etc.
Two way ANOVA
1 measurement variable and 2 nominal variables
Ex: measure response to 3 different drugs in men and women. Drug treatment is one factor and gender is the other.
Post Hoc analyses
ANOVA only tells us that the smallest and largest means likely differ from each other. What about other means?
- run a post hoc test!
Only used if Ho is rejected.
Mann-Whitney U Test
Non-parametric alternative to two-sample t-test
- Uses rank of measurement instead of actual measurement
- Calculate and look up value in table. If calc
Pearson Correlation Coefficient (r)
Measure of the linear correlation between two variables.
1 - positive correlation
0 - no correlation
-1 - negative correlation
Linear Regression
Goal is to create a line that minimizes the sum of the squares of the vertical distances of the points from the line.
It assumes that your data is linear, and finds the slope and intercept that make a straight line that best fits your data.
Categorical Data
No mean, median, mode, or normal distribution.
Data divided into groups (Yes/no)
Discrete data? (nominal, ordinal?)
Contingency Table
To measure associations bw categorical variables
Cat 1 C a b a t c d 2
- Assumes that all data are independent (each person fits into one box only)
- Can be any size
Chi Square
- Measures the observed frequencies and compares them to the expected.
How to calculate:
1. Expected measures in each box are calculated
- (Total in row * total in column) / TOTAL
2. Calculate Chi Square
X2 = Sum (obs - exp)^2 / exp
3. Compare to table
- Calculated X2 should be > critical value to reject Ho
Fischer’s Exact
X2 is not valid for 2x2 contingency tables with very small samples. Use Fischer’s exact
Chi Square Assumptions
- Data are frequency data
- There is an adequate sample size
- Measures are independent of each other
Odds Ratio
Odds - Probability of the event occurring compared with the probability that it will not occur.
Odds ratio is the ratio of 2 odds. Used mostly in case-control studies. Measure of association between an exposure and an outcome.
OR = (a/c)/(b/d) = ad/bc
= odds that a case is exposed/ odds that control is exposed
OR = 1 (No difference in odds of exposure)
OR > 1 (Increased odds of exposure)
OR
RCT
Bias
- Randomization removes selection bias
- Investigator bias (use blinding)
- Subject bias (use blinding)
Things to control:
- Diet
- Health changes
- Non compliance
- Drop outs
- Lifestyle
- Events
Investigator Bias
Allocation concealment: ppl randomizing individuals are blinded as to which subjects go into which group
Investigator blinding
Subject Bias
Hawthorne Effect: People change behavior in a study
Subject blinding
Case Control Study
- Compare patients who have a disease with patients who do not, and look back retrospectively to compare how frequently the exposure to a risk factor is present
Threats to internal validity:
- Control group selection (matching)
- Recall bias
- Can’t determine risk directly (use odds ratio)
–> look at people who already have disease and determine odds of exposure. What are the odds that the diseased group was exposed?
Cohort Study
A cohort is a group of people who share a common characteristic or experience within a defined period. Follows the cohort over time and the outcomes are compared to a subset of the group who were not exposed to the intervention.
- Incidence studies
–> measure how many people develop disease out of a total. What is the relative incidence of disease in both groups?
Risk
= number subjects with unfavorable event in arm/total number subjects in arm
(Absolute risk)
–> makes risks/benefits look SMALLER
Relative Risk (RR)
risk in treatment/risk in control
(Exposed/nonexposed)
RR = [a/(a+b)] / [c/(c+d)]
R = 1 : no difference in risk
R 1 : more events in tx group vs control
–> makes risks/benefits look BIGGER
Exposure
Can occur at a single point in time or over a period of time.
Characterizing exposure:
- Ever been exposed
- Current dose
- Largest dose taken
- Total cumulative dose
- Years of exposure
Attributable Risk
The additional incidence of disease related to exposure, taking into account the background incidence of disease from other causes.
- Implies that the risk factor is a cause and not just a marker
Also called RISK DIFFERENCE - difference bw 2 absolute risks
ARR = Risk control - Risk treatment
Relative Risk Reduction (RRR)
By how much the treatment reduced the risk of bad outcomes relative to the control group.
RRR = 1 - RR
= [Risk control - Risk treatment] / risk control
NNT
NNT = 1/ARR (in decimal) OR 100/ARR (in percent)
Sensitivity
Probability of testing positive, given patient has disease
= a/(a+c)
Overall Accuracy = (a+d)/(a+b+c+d)
Specificity
Probability of testing negative, given that patient does not have disease
= d/(b+d)
Overall Accuracy = (a+d)/(a+b+c+d)
Prevalence
Proportion of a group of people possessing a clinical condition or outcome at a given point in time.
= (a+c)/(a+b+c+d)
Other names
- prior probability
- pretest probability
Positive Predictive Value
Probability of having dz, given a positive test
= a/(a+b)
Negative Predictive Value
Probability of not having dz, given negative test
= d/(c+d)
Likelihood Ratio
Positive
LR+ = Sensitivity / (1 - Specificity)
Negative
LR- = (1 - Sensitivity) / Specificity
The probability of that test result in people with the disease divided by probability of the result in people without the disease.
Ratio expresses how many time more (or less) likely a test result is to be found in diseased, compared with non-diseased, people.
Posttest Odds
Pretest Odds x Likelihood ratio
The 4As of EBM
Ask
Acquire
Appraise
Apply
New EBM pyramid
Systems (Not in effect--> put sx in computer) Summaries (Up to date) Synopses of Syntheses (Dynamed) Syntheses (Systematic reviews) Synopses of Studies (Journal club) ------- Individual studies (primary studies)
Berkson’s Bias
If the sample had been taken from a hospitalized population.
- Systematically higher exposure rate among hospital patients, distorting odds ratio, etc.
Review Article
- Synthesize results and pull together major findings
Strength
- Provide good discussion from experts
Weakness
- Subject to bias of author
Systematic Review
Reviews of the literature that follow a prescribed protocol to remove bias
Goals
- Provide up to date summary of all good published lit
- Assimilate large amounts of data
- Objective collation of results
- Reliable recs
Elements of a Systematic Review
- Define a specific question (PICO)
- Find all relevant studies (pub and unpub)
- Inc sensitivity and reduce bias
- Select strongest studies
- RCTs/no obs?
- Describe scientific strength of selected studies
- Determine if quality is assc with results
- (4&5) examine internal validity
- Review for bias
- Review includes > 1 researcher
- Summarize studies in figures (forest plots)
- Determine if pooling of studies (meta analysis) is good
- If yes, calculate summary effect size and CI
- Identify reasons for heterogenity if present
Meta Analyses
- If results are similar, can be pooled and analyzed together
- Requires study question similarity
Results weighted by sample size
- Fixed effects model: when studies ask the same Q
- Random effect model: assumes studies are asking diff Qs but somewhat similar
Systematic review Sources of Bias
Study bias
- Biased samples
- Berkson’s bias
- Subject/investigator bias
- Author bias
- > 1 researcher to prevent bias
- Publisher bias
Web of Causation
- To counter reliance on single - cause model
- Includes biological, behavioral, and social factors
- Graphical depiction
Cause - Sufficient
With it the effect will result regardless of the presence or absence of other factors
Cause - Necessary
- Without it, the effect will not occur
- Need other factors for the event to occur
Single Cause Model
Koch’s Postulates
Bradford Hill Criteria
- Temporality
- Exposure precedes dz
- ONLY required criteria on the list for causality - Strength of association
- Strong ass doesnt ALWAYS –> causality
- Week ass doesnt negate - Dose response
- lack doesn’t negate - Reversibility
- lack doesn’t negate - Consistency with other knowledge
- Consistent results across study designs - Biological plausibility
- lack doesn’t negate - Specificity of the associaiton
- exposure ass with one specific dz outcome
- weakest of criteria
- many dz have multiple causes - Analogy
- cause is analogous to other established relationships
(9. Coherence)
The more criteria fulfilled, the stronger the case for causality.
Ecological Studies
Level of analysis is groups rather than individuals
- Aggregate studies
Fallacy
- Ascribing group characteristics to individual members of that group
- Assumption only valid if exposure is homogenous within groups
- We don’t know whether the individuals had high rates
- All we have is average values of smoking levels and rates of lung ca mortality in each country
Components of a Medical Paper
- Abstract
- Introduction
- Study design, subjects, sampling, sample size, outcome, stats - Methods
- Results
- Discussion
- References
Cross Sectional Study
A study that examines the relationship between diseases and other variables of interest at a single point in time.
Determine your sample first and see what exists in that sample.
Prevalence
Peer Review Process
- Author submits manuscript
- Editor reviews
- Rejects manuscript
- Assigns to reviewers for external eval - Reviewers review
- Make recs to editor - Editor makes decision
- Rejection, modification, publication - Publisher publishes finalized manuscript
Levels of Evidence
- Systematic review of randomized trials
- Randomized trials
- Non-randomized controlled cohort/follow-up study
- Case-series, case-control, or historically controlled studies
- Mechanism-based reasoning
How to critically appraise a medical paper?
- Choose appropriate tool
- Answer all questions in the tool
- Draw your own conclusions
- May stop if the quality of a paper is of concern
- RWJMS general article review sheet
- Oxford centre for EBM
- EQUATOR network
How to Report trials???
CONSORT - Consolidated standards of Reporting Trials
- 25 item checklist
- A flow diagram
OR???
The equator network to report things.
