Epidemiology & Biostatistics

• values that correspond to unordered categories or classes
• eg. hair color, gender, state of birth, etc.
• have no ordering & are summarized as percentages
• nominal data that can take 1 of only 2 values (eg. having or not having gonorrhea are called binary/dichotomous data)
• type of Categorical data

• when the categories of a variable have an inherent order to them
• eg. Glasgow Coma Score, severity of adverse events (AE), “satisfaction”
• type of Categorical data

• data restricted to specific integer values
• eg. number of children, bacteria count
• type of Measured data

• unrestricted real numbers
• eg. weight, age, degree of stenosis, physician salary
• type of Measured data

• age itself can be reported using mean & standard deviation (SD)
• OR it can be grouped into categories - 45-54 yr, 55-64, ≥ 65 - & reported as percentage (x % were at this age, etc.)

• useful for summarizing Categorical (nominal & ordinal) data
• displays # or % of individuals in each of the categories
• their bars do NOT touch to emphasize that the categories are DISTINCT from each other & MUTUALLY EXCLUSIVE

• bars touch each other to emphasize the data are CONTINUOUS (measured)
• like in a bar chart, each bar of a histogram displays the # or % of individuals in each interval
• used to display the distribution of data in a sample
• is VERY good at displaying extreme data points

y-axis = frequency (or %) - aka # of individuals whose cholesterol falls between 5 & 6

• sum of observations divided by the # of observations
• if one were to place “weight” on a numeric scale, the arithmetic mean would be at the balancing point
• because of this, it’s SENSITIVE to EXTREME observations (outliers)
• 

• NOT sensitive to extreme observations
• the value that divides an ordered data set in half
• 50% of the data are ABOVE the median value, & 50% are below it

refers to the value where 25% of the data are below it

the value where 75% of the data are below it

• 25% of patients have an LVOT below 7 mmHg
• 50% of patients have an LVOT below 16 mmHg
• 75% of patients have an LVOT below 26 mmHg
• the interquartile range (7-26) contains 50% of the observations

• an illustrative tool for displaying the MEDIAN as well as the 1st & 3rd quartiles
• the Whiskers extend to the edge of extreme values in the data set
• more extreme values are individually displayed as dots
• can be useful when comparing 2 or more groups

• even though women & men have similar distributions of cholesterol levels by age group, it’s easy to see that the median cholesterol levels increase as women age
• however levels remain relatively stable in men as they age

when looking at distributions of data (using either histograms or box plots), it’s important to characterize them in regard to how far a distribution deviates from symmetry

• when the mean, median, & mode have identical values
• normal distributions are symmetric
• 

• mean the skew is “to the right” - this manifests in a graph with a tail on the right side of the graph (is more “pronounced)
• mean > median
• 

• the skew is to the left
• the left “tail” is more pronounced
• mean < median
• Mode
• the most common (repeated) value in a dataset
• 

• Positive/Right Skew: both the right whisker & tail are ELONGATED
• Negative/Left Skew: both the left whisker & tail are ELONGATED
• Symmetrical: both whiskers & tails are equal distribution from the center
• Extremes: box plot displays them as individual data points; histogram displays them separately, with discontinuity
• 

• numerically characterizes the amount of variability (data dispersion, data spread) among observed data points
• the mathematical formula calculates the average distance from the mean
• square root of the variance

• indicate how many SDs an observation is above or below the mean of a distribution
• computed by subtracting the mean from the observation & dividing the the SD
• Z = random variable - the mean / SD
• or (x - μ) / σ

there are none because the mean & SD have the same units, the units cancel out

• 68
• 95
• 99
• ~ 68% of the observations lie within ±1 SD of the mean
• ~95% of samples lie within ±2 SDs of the mean
• ~99% lie within ±3 SDs of the mean
• 
• someone with a Z score of 1 is 1 SD away from the mean

a probability expressed as a percentage

all individuals we’re interested in studying

a representative subset chosen from said population of individuals we’re interested in studying

• suppose a population has a mean = μ & SD = σ
• 
• from the population, you can take many samples, eg. of size 10 (n = 10), & find the means for each sample
• [the distribution of observations in each sample will be different from each other]
• 
• can continue the process of selecting random samples an infinite # of times
• if you were to plot all the calculated means they would likely show a normal bell-shaped curve
• 
• the population mean would be the center of the curve, & the variability among the infinite # of means = the STANDARD ERROR

• SE = pop. standard deviation / √ sample size
• SE = σ / √n (eg. 41/√10)

describes the entire spectrum of sample means that could occur for all possible random samples of a given size n from means that are COMMON (near the center of the distribution/curve) to means that are RARE (near the edges of the distribution)

• given the existence of a population mean (μ) & population standard deviation (σ) from any distribution of any shape, the CLT states that the distribution of sample means (computed from random samples of size n from a population) will have its mean centered at the population mean
• the standard deviation of all the means = standard error
• if n > 30, the shape of the sampling distribution will be ~normal
• important because it forms the basis of all statistical tests

• used to approximate the population parameter of interest using a point estimate (eg. sample mean)
• a range of values that, with a known level of certainty, includes (surrounds, captures) the unknown population value

that we are 95% confident that the range of values between the lower & upper limits contains the true population value

• point estimate ± critical value * SE(point estimate)
• critical value indicates a level of certainty (eg. 95%)

1. We are 95% confident that _______ levels (scores, values, etc.) ranging from x to y (units) capture the true mean _________.

2. We are 95% certain that the true mean ________ level is between x to y (units).

1. Increasing the level of confidence (eg. 95% → 99%, you widen the CI to be more certain in capturing the true population value)

2. More variability among the observations (eg. larger standard error); more variability implies less precision in capturing the true pop. value

3. a smaller sample size (there’s less information involved in capturing the true population value)

specifying the null & alternative hypothesis

null: statistical statement that indicates no difference, response, or change exists

alternative: contradicts the null

• significance level of the test
• usually set at 5%
• represents the threshold beyond which the null hypothesis would be rejected in favor of the alternative hypothesis

• probability of obtaining a test-statistic at LEAST as extreme as the one that was actually observed, assuming that the null hypothesis is true
• can be interpreted as the likelihood that the observed result could’ve occurred by chance alone

• ratio of the observed mean difference to the amount of sampling variability (given via standard error)
• t values close to 0: support the null hypothesis
• t values further from 0: support the alternative hypothesis
• eg. T = 3.4 implies the mean difference is 3.4 standard errors ABOVE 0 (supports the alternative hypothesis)

• it is the area under the sampling distribution of mean differences in both the L & R tails
• the areas that yield the p-value are calculated from a t distribution
• half the p-value comes from the L tail & half comes from the R tail

it means that the alternative hypothesis is true & there’s a 0.07% chance that the difference could have occurred by chance alone

1:1

whenever the CI doesn’t contain the parameter of interest, the null hypothesis can be rejected

eg. if a 95% CI about a mean difference EXCLUDES 0, we can conclude that the mean difference is significant at an alpha level of 5%

however, if a 95% CI about a mean difference INCLUDES 0, we cannot reject the null hypothesis

• used when the outcome & exposure variables are categorical
• eg. % of current smokers among newly diagnosed diabetes patients
• the basis of the chi-square test is to quantify the extent of agreement between the observed results gathered from data collection & the EXPECTED results one would observe if the null hypothesis were true

• chi square values near 0: accept the null hypothesis
• large chi square values: reject the null hypothesis, accept the alternative hypothesis
• χ2 = 0.61 implies good agreement between observed & expected results → accept the null hyp.
• the p-value for a chi square test is calculated as a tail area under the chi square distribution

• when the null hypothesis is rejected (alternative hypothesis accepted) when it shouldn’t have been
• a difference in the sample is observed when there is actually no difference in the population
• (guilty verdict when defendant is innocent)

• when the null hypothesis is not rejected (alternative hypothesis is rejected) when it should have been rejected
• (not guilty verdict when defendant is guilty)

• 1 - β
• the probability that a statistical test will RESULT in the REJECTION of the null hypothesis (acceptance of the alternative hypothesis) when it SHOULD be rejected
• (when a jury correctly assigns guilt)

• 1. when a study is being planned, to determine the # of participants to enroll
• 2. when the null hypothesis is accepted (NOT rejected)

• 1. ↑ the expected effect size (eg. expected association, difference in means)
• 2. ↓ the expected standard error by:
• ↑ the sample size or ↓ standard deviation

• a Type I error is made
• this is because the significance level for the test is pre- specified
• multiple testing inflates the overall Type I error rate

• 1 - (1 - α)^# of tests
• eg. if 5 tests are performed at α = .05 (5% sig level)
• error rate = 1–(1–0.05)^5 = 0.226 (22.6%)
• this means there’s a 22.6% chance of finding at least 1 significant difference when it doesn’t exist
• (instead of the usual 5% chance of a false-positive finding)

• 1. Primary
• 2. Secondary
• 3. Tertiary

• screening is done to prevent the disease
• eg. serum lipids are screened to prevent coronary artery disease

• the attempt to reduce the impact of the disease
• eg. mamography

• aims to improve the quality of life of those with the disease
• eg. metabolic bone screen for pathologic fractures

those that are serious, common, & ones that would benefit from treatment before symptoms/sign’s develop

making sure it’s available, inexpensive, low risk, easily performed, reliable (it’s results can be reproduced), & accurate (results are correct)

• people with a disease: true positives & false negatives
• people with no disease: true negatives & false positives

• 1. Sensitivity
• 2. Specificity
• these are measures of test performance

• true positives / true positives + false negatives
• calculated ONLY among individuals WITH the disease
• given the disease is present, the likelihood of testing positive
• TP / TP + FN

• true negatives / true negatives + false positives
• calculated only among individuals WITHOUT the disease
• given the disease is not present, the likelihood of testing negative
• TN / TN + FP

• TP / TP + FP
• calculated ONLY among individuals that test positive
• the number of true positives divided by the sum of true & false positives

• TN / TN + FN
• calculated only among individuals that test NEGATIVE
• the number of true negatives divided by the sum of true & false negatives

because changes in prevalence can alter Predictive Value Positive & Predictive Value Negative

• predictive value positive increases
• predictive value negative decreases

• predictive value positive decreases
• predictive value negative increases

they remain unaffected by prevalence!

a method that quantifies the likelihood that a given test result represents true disease or not

it’s the ratio of the chance that a certain test result will be found in a patient who has the disease versus the chance that the test result will be found in a patient who does not have the disease

LR for a positive result = sensitivity / (1-specificity)

LR for a negative test result = (1-sensitivity) / specificity

• to calculate LR in general:
• # of people with a disease with a certain test outcome / # of people without a disease with the same test outcome

• volunteers are randomized to:
• 1. experimental arm
• 2. placebo/standard/control arm
• both groups are followed over a certain period of time
• then the incidence of outcome in both groups is measured & compared
• 

• 1. Enroll volunteers using strict inclusion/exclusion criteria
• 2. Allocate to treatment groups by randomization
• 3. Follow for relevant time period
• 4. Ascertain endpoint/outcome
• 5. Analyze results

Randomization

• reduces potential bias by:
• removing investigator treatment preference, volunteer treatment preference, & balances the trial arms
• it results in similar risk profiles in both groups

• investigators & volunteers don’t know whether a person received a treatment or not
• this prevents investigators & volunteers from biasing the results
• treatments should look alike (can be difficult if it’s say radiation v. chemotherapy) however the outcome/end points committee can be blind when they evaluate both groups

• it removes the “placebo effect” from the measure of treatment effect
• it removes volunteer influence by blinding
• it removes investigator influence by blinding
• these are especially important if the outcome is SUBJECTIVE

• long enough to detect differences in outcome as well as differences in side effects
• often side effects aren’t encompassed in the relative time period
• while longer follow up might be beneficial in terms of detecting side effects, it can also lead to a potentially larger loss to follow up
• this loss-to-follow up is a bias

• those evaluating should be blind to treatment assignment
• a PRECISE definition should be used to define the outcome

• volunteers analyzed in the group to which they were randomized regardless of actual treatment received (eg. if they’re lost to follow up, don’t comply, etc.)
• “best” measure to be used to analyze the results
• the idea is to retain benefits of randomization

compare the two arms using Relative Risk (RR)

• Outcome Incidence in Experimental Group / Outcome Incidence in Control Group
• for a cohort study, RR can be written as:
• Outcome Incidence in Exposed / Outcome Incidence in Unexposed Group
• a ratio of risks
• 

• Outcome Incidence in Experimental = Outcome Incidence in Controls
• RR = 1 is also called the “Null Value”

• when the outcome incidence is higher in controls than in the experimental group
• the denominator (controls) is bigger, so RR < 1

• when the outcome incidence is higher in the experimental group than in controls
• the numerator (experimental group) is bigger, so RR > 1

• they ascertain effect modification
• eg. is the effect of a treatment on an outcome the SAME in all people that participated in the trial (eg. in women & men, young & old, etc.)
• subgroup analyses are important to clinicians because they may identify subgroups in which treatment is helpful or harmful

• 1. Observational
• 2. Interventional

studies within these categories can either be Descriptive or Analytic

• in case series there is NO comparison group
• in cross-sectional studies there’s an experimental & control group

• Case Report
• Case Series
• Cross-sectional
• Correlational

• Cohort Studies
• Case-control Studies

• Case Report
• Case Series
• these 2 kinds of studies can ALSO be categorized as interventional because the exposure can be chosen by an investigator

Randomized Control Trial

• observational studies that are analytic (as opposed to descriptive) in nature
• investigator recruits 2 types of individuals: exposed & unexposed
• the investigator then follows these 2 groups through time & eventually measures the incidence of outcome in each
• eg. twin studies comparing twins who had different exposure levels
• 

the investigator starts a study (TODAY) & follows exposed & unexposed volunteers through time (eg. for 10 years FORWARD) & then compares the incidence of outcome in exposed v. unexposed volunteers

• the investigator again might start the study today, but then looks BACK in time with the assistance of medical records to determine who’s exposed & who’s unexposed
• the investigator might then again compare current day outcomes in both groups

• when you’re interested in incidence rates or predictors MORE than the effects of interventions
• it can be used before a randomized trial is proposed (eg. to generate hypotheses like the effects of hormone replacement therapy or dietary fat)
• when exposure CANNOT be randomized (eg. genes, race, BMI, serum cholesterol, potentially harmful exposures [cigarettes, drugs, pesticides])

• a Cohort Study has a LOWER validity than an RCT
• this is mostly because there is no randomization in a Cohort Study
• it’s also more difficult to measure exposure because often we rely on self report

• to have an ACCURATE measure of true exposure
• definition of Exposure should be clear & concise
• it should be measured with accurate instruments, & the same method/instrument should be used in the exposed & unexposed
• Exposure should be assessed BLIND to outcome (to avoid investigator bias)

• to have (again) an ACCURATE measure of the true outcome
• a clear & precise definition is needed
• outcome should be measured with an accurate instrument
• potential sources of outcome data include disease registries, medical records, death certificates

• they work well for exposures that can’t be randomized (genes, drug use)
• they’re good for RARE exposures
• they can assess multiple outcomes
• they can generate Incidence data

• they’re good for studying diseases with long latency
• they take less time to complete
• are less expensive
• (less time & resource intensive)

• the exposure can’t be randomized
• they’re bad for rare OUTCOMES (because you have to follow many many people over a long time to see the outcome)
• - this long follow-up time to observe outcomes can also lead to loss to follow-up & a change in someone’s exposure status
• they can be EXPENSIVE because of the number of years needed to follow the cohorts
• subjects need to be free of the outcome at the start of the study & sometimes that can be hard if diagnosis can’t easily be done or isn’t clear-cut
• in a retrospective cohort study, the data may not be available or adequate

• can be used as a measure of association between exposure & outcome
• the probability (risk) of developing the disease if exposed compared to the probability of the developing disease if unexposed
• RR = Incidence of Disease if Exposed / Incidence of Disease if Unexposed
• unexposed (controls) are in the denominator

• exposure promotes the outcome
• eg. a larger # of people got the disease (outcome) if they were exposed
• larger numerator

• exposure prevents outcome
• eg. a larger # of people DIDN’T the disease (outcome) if they weren’t exposed
• larger denominator

• the exposure had no effect of on the outcome
• the same proportion of people got the disease (outcome) in both the exposed & unexposed groups
• RR = 1 is the null value

• Incidence in Exposed - Incidence in Unexposed
• it’s the incidence DUE to exposure, because it’s the difference between the two incidences
• it compares the incidence of disease (outcome) in the exposed group & the incidence of outcome in the unexposed group
• is another measure of association between exposure & outcome of interest
• 

1 / Attributable or Excess Risk

this is the # needed to treat to prevent 1 occurence of a disease (eg. stroke)

• RR can be useful in helping an individual understand their personal risk of disease by continuing to expose themselves to an exposure
• RR might be what’s shared with a patient
• AR is important in public health advocacy work (eg. when talking with legislators or policy makers)
• AR helps you understand the # of cases that can be averted, or can be converted into the cost of a disease in a population

• the distortion of a study’s results
• the results of the study don’t reflect the truth when there’s systematic error in the measurement of an association between 2 variables

• Confounding
• Selection Bias
• Information Bias (random + non-random)
• Loss to Follow-up
• *unlike with confounding, the other sources of bias CAN’T be corrected during analysis & MUST be avoided in the study design
• 

any misclassification of the exposure that’s the SAME in both outcome groups

• 1. ALL volunteers lie about substance abuse → underestimate of exposure
• 2. ALL volunteers over-estimate daily physical activity → overestimate of exposure
• 3. ALL subjects have trouble recalling average red meat consumption
• with these 3 examples, the bias is ALWAYS toward the NULL
• there’s a “watering down” of the association

any misclassification of the outcome that’s the SAME in both exposure groups

• 1. an investigator may UNDER-diagnose outcomes in ALL volunteers (fewer people will have the outcome of interest)
• 2. 1. an investigator may OVER-diagnose outcomes in ALL volunteers (more people will have the outcome of interest than should)
• 3. a disease is difficult to conclusively diagnose (eg. MI)
• again, the bias is ALWAYS toward the NULL (there’s a watering-down effect)

any misclassification of the exposure that’s DIFFERENT in outcome groups

• 1. mother’s with babies born with FAS LIE about alcohol use
• 2. volunteers who haven’t had an MI over-estimate physical activity
• 3. investigators over-estimate exposure in people with a disease & under-estimate exposure in people without a disease
• bias can either be toward OR away from the NULL

any misclassification of the outcome that’s DIFFERENT in exposure groups

• 1. investigators under-diagnosing an outcome in people who’ve had surgery v. people who haven’t had surgery
• 2. investigators over-diagnose a disease (outcome) in those who were exposed v. people who weren’t exposed
• 3. investigators UNDER-diagnose people in the exposed group & OVER-diagnose people in the unexposed group
• can lead to bias toward or away from the null

by BLINDING investigators to both the exposure & outcome of interest

the ability to apply the findings from our study population to a larger population

• the degree to which a study’s findings represent a true reflection of the exposure-outcome association in the population
• how close the estimated relative risk in a study is to the true (but unknown) relative risk
• internal validity = absence of bias (bias = “distance from the truth”)

• 1. Rule out confounding
• 2. Rule out other sources of bias
• 3. Rule out chance with statistical tests

double-blind RCTs are considered the “gold standard” of epidemiologic study designs because the internal validity (absence of bias) is greater than in other study designs

1. it must be a risk factor for / associated with the outcome

2. it must be associated with the exposure, or unbalanced in the exposure group

3. it must NOT be on the intermediate path between the exposure & outcome, aka it can’t be a mediator on the causal pathway between the exposure & outcome

researchers showed that menopause increased a woman’s risk for heart disease



what was actually happening was that age was acting as a confounder, creating the ILLUSION of a positive association between menopause & CHD

the criteria: age IS a risk factor for CHD, age is UNBALANCED across exposure groups (pre/postmenopausal women tend to be different ages), & age is NOT on the causal pathway (like endogenous estrogen, which would be)

• an attribute or exposure associated with an increased or decreased probability of a health-related outcome
• not necessarily a direct cause of disease
• eg. vaccines are an example of a “risk factor” that may prevent disease.
• also called exposure, predictor, or determinant

• you can Randomize individuals into the 2 arms of the study
• you can Restrict which subjects you include (eg. if sex is a cofounder, just do the study in men)
• you can MATCH subjects on specific characteristics that you KNOW to be confounders (eg. sex, age, race, ethnicity)

via Stratified analysis & Multivariable analysis

• a way to avoid confounding in the study DESIGN
• eg. TWIN studies
• if sex, ethnicity, & age were confounders, then matching eliminates their ability to confound

a factor OTHER than the exposure of interest (or even the disease) that can modify the exposure-disease association

Conf: look at a difference between crude & adjusted RR/OR (eg. within the same variable but different when adjusted for various other variables)

EM: look across subgroups (strata); applies to variable of interest (eg. age)

• visually summarizes the relationship between two continuous variables
• eg. vitamin D serum levels v. vitamin D supplementation

• RR & OR are single numbers used to quantify the relationship between 2 binary variables
• when you have 2 continuous variables, as in this example, the relationship can also be quantified by a single number: r

• quantifies the magnitude & direction of LINEAR relationships between 2 CONTINUOUS variables
• is the average of the product of the Z-scores between the two variables of interest
• has NO units
• range = - 1 → +1

• 0: no linear relationship
• +1: a perfect positive linear relationship (positive means as the values of 1 variable increase, so do the values of the other variable)
• -1: a perfect inverse linear relationship (negative means as the values of 1 variable increase, the values of the other variable decrease)
• 

a statistical tool for evaluating the relationship of one or MORE independent variable (predictors) to a SINGLE continuous dependent (outcome) variable

in addition to evaluating relationships, regression analysis can also be used to predict outcomes (via deriving prediction equations)

• used to fit a straight line (describe & predict a linear relationship) through points on a Scattergram
• done in relation to 2 variables (X & Y axis)
• the straight line is derived mathematically so the same # of data points lie above & below the line

• Y = β_o + β₁X
• Y: the predicted value (outcome)
• B_o: the INTERCEPT/cOrrelation coefficient
• B₁: the s1ope coefficient

for every 1 unit of supplemental vitamin D taken, the estimated increase in vitamin D serum levels is 0.0249 units

Y = 64.7 + 0.0249X

1. the s1ope coefficient β₁ has UNITS, the cOrrelation coefficient doesn’t

2. the cOrrelation coefficient β_o just assesses the strength of the relationship; it doesn’t describe how the dependent variable changes in relation to changes in the independent variable

a 1 to 1 relationship

• proportion (percentage) of variation in the outcome variable that can be explained by the exposure
• ranges from 0 → 1 (0 → 100%)

eg. if R² = 0.73, then 73% of the variability in the outcome can be explained by the exposure

used to describe & predict a linear relationship between 1 dependent variable & 2 or more independent variables

the adjusted r² corresponds to one from a multiple linear regression, & sometimes explains more variability (the value increases)

Y = β_o + β₁X₁ + β₂X₂ + β₃X₃…

it can assess multiple exposures

it can assess potential confounding

• the β values correspond to each variable - so for every 1 unit that the independent variable changes by, the dependent variable (what you’re measuring) changes by the β coefficient that corresponds to that specific independent variable
• this β coefficient exists after adjusting for all variables in the regression model
• 
• vitamin D eg.: for every 1 ounce of fish eaten, vitamin D serum levels increase by 2.01 (the individual β coefficient for the variable “fish intake”)

• when the outcome is binary (yes/no, present/absent - is an extension of the 2 by 2 table)
• can be used to calculate adjusted odds ratios & relative risks for more than 1 extraneous variable
• to generate prediction equations in the case of binary outcomes

• π(x) = e^β₁+β₁x / 1+ e^β₁+β₁x
• β_o & β₁: unknown parameters
• x: exposure
• π: proportion of the outcome (1=yes, 0=no)

as a non-linear equation to ensure the outcome is bounded between 0 & 1

if logistic regression is used to predict the risk of outcome, the risk estimates can’t be less than 0%

• convert the log equation to a linear one via a logit transform
• ln[odds] = ln [π(x) / 1 - π(x)] = β_o + β₁

when the time to an event is AS important as whether the event occurs or not

• accommodates varying lengths of follow-up
• outcome of interest is the TIME until an event occurs rather than whether it does or doesn’t occur

• 1. staggered entry into the study
• 2. subjects might drop out of the study, are lost to follow up, or may die
• 3. subjects who don’t have an outcome by the end of the study

• 
• F: saw physician (outcome achieved)
• L: lost to follow up at noted time
• C: censored (outcome not achieved in specified time period)
• question to ask: What is the rate (risk, chance, likelihood) of being seen by a physician (outcome) within __ minutes of arriving at the doctor’s office?

• # of new cases in a specified time period / total number of units of person-time
• a simple but MORE precise measure of incidence than cumulative incidence
• the advantage of ID is that it accounts for unequal follow up & loss to follow up

• 12 total patients
• 3 patients were seen within 15 minutes of arrival
• minutes waited by everyone = 146
• ID = 3 / 146 = 0.0205
• 0.0205 * 100 = 2.05 patients per 100 patient-minutes
• 2.05*15 = 30.8 patients will be seen per 100 patient-minutes within 15 minutes

there’s a 30.8% CHANCE a patient will be seen by a physician within 15 minutes of their arrival

• an approach to display the cumulative proportion of participants who did NOT experience an outcome event over time
• each step plotted on a K-M curve represents an outcome event
• because K-M analysis & corresponding curve accounts for varying lengths of follow up, the K-M estimates are VERY close to Incidence Density

• used to compare Kaplan-Meier curves between 2 or more groups
• the comparrison is made across the entire event free survival curve & not at any particular period of time

does the same thing that a logistic regression does (accommodating multiple exposure variables [including potential confounders & effect modifiers] for time to event data

• h(t) = h0(t)e^{β₁x₁ + β₂x₂…}
• h(t): hazard @ time t

• Logistic Regression is an extension of the chi-square test
• Cox Regression is an extension of the event free survival analysis
• while the log regression model yields odds ratios, a Cox regression model yields hazard ratios

• as Relative Risks
• hazard ratios are byproducts of the Cox regression model

• a type of observational analytic study in which subjects are selected based on their disease status
• subjects are classified as Cases (having the disease) or Controls (not having the disease)
• cases are identified by the outcome/disease being clearly & precisely defined
• Good for rare OUTCOMES (diseases); Bad for rare exposures

• investigators should be blind to a person’s exposure status when assessing if the person is a case
• similarly, exposure should be assessed blind to outcome

prevalent (old + new) cases may be survivors & therefore may not be representative of “typical” cases

• INCIDENCE, because we started with cases (people who already have the disease of interest)
• therefore relative risk can’t be calculated either

Odds Ratio

OR is a good estimate of the relative risk when the disease of interest is rare (<10%)

• in both a prospective & retrospective cohort study, disease incidence can be calculated → RR can be calculated (as can OR)
• Case-control can ONLY use OR

odds that a case was exposed / odds that control was exposed

• Good for rare diseases
• Allow for evaluation of multiple exposures
• Efficient (re: time & cost)
• Avoid potential ethical issues of an RCT (eg. can’t randomize subjects to a harmful exposure like smoking)

• they’re BAD for rare exposures
• Comparability of cases and controls might not be given
• Can’t generate incidence data because time isn’t known
• there’s potential Selection bias, Interviewer bias & Recall bias (are all controls really free of the disease?)

• caused by how the study subjects were selected for the study
• results in the association between the exposure & outcome not being representative of the target population’s true association
• to avoid selection bias, the exposure & outcome shouldn’t both be mentioned in recruitment material

• RCT
• because subjects join the study before they know their exposure status & because the outcome has not yet occurred

• Cases (who have experienced an adverse health outcome) may be more likely to recall exposure histories than controls
• eg. cancer cases recall pesticide exposure more readily than people without cancer
• it can attenuate association between disease & exposure towards the null or exaggerate association away from the null
• a differential (non-random) misclassification of the EXPOSURE

• Retrospective Studies only
• because outcome has already occurred when the Exposure is assessed

• 
• Case-control: recruit people based on their disease (outcome) for the outcome arm then find controls to match that don’t have the disease
• Cohort: observational study, recruit people just based on whether or not they have or don’t have a certain exposure → outcome is assessed “later”

because there’s NO randomization

[small RCTs are more subject to confounding than large RCTs]