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\bs
\begin{center}
\begin{Large}
\blue Measures of Frequency and Effect in Clinical Research \\
\end{Large}
\begin{large}
\vspace*{.5in}
Jonathan S. Schildcrout, Ph.D. \\
Department of Biostatistics \\
Vanderbilt University School of Medicine \\
\vspace*{.25in}
November 18, 2004
\end{large}
\end{center}
\es
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\bs
\slidetitle{Overview of this Session}
\begin{footnotesize}
\begin{itemize}
\item Study Design
\begin{itemize}
\item Clinical trial
\item Cohort study
\item Case-control study
\end{itemize}
\item Frequency measures
\begin{itemize}
\item Incidence
\item Prevalence
\end{itemize}
\item Measures of Association and Effect
\begin{itemize}
\item Attributable Risk
\item Relative Risk
\item Odds Ratio
\item Correlation
\item Analysis of paired data (measure of change)
\end{itemize}
\end{itemize}
\end{footnotesize}
\es
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\bs
\slidetitle{Study Design}
\begin{small}
\begin{itemize}
\item {\blue Descriptive Studies (a.k.a., hypothesis screening studies)}
\begin{itemize}
\item Used to study variation in disease frequency by demographic characteristics, place and time
\item Lack a hypothesis specified in advance $\Rightarrow$ hypothesis generating
\item Relatively low cost studies using pre-existing data
\end{itemize}
\item {\blue Analytic studies}
\begin{itemize}
\item Researcher has a pre-specified hypothesis in mind
% \item In most cases causality is difficult to assert
\item Experimental vs. observational
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Study Design: Analytic Studies}
\begin{small}
\begin{itemize}
\item {\blue Experimental studies}
\begin{itemize}
\item An experiment is a set of observations conducted under controlled conditions, where the researcher manipulates conditions to ascertain what effect the manipulation will have.
\item In general, experimental studies are those in which the researchers manipulate exposure.
\end{itemize}
\item {\blue Observational or non-experimental studies}
\begin{itemize}
\item A study that does not involve intervention
\item Observe natural course of events where changes in one characterstic is studied in association with changes in other characterstics
\item Often necessary when unethical or infeasible to manipulate exposure
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Study Design Taxonomy}
\begin{footnotesize}
\psmatrix[colsep=.5cm, rowsep=.2cm]
& & & \fbox{Randomized Clinical Trial} \\
& & \fbox{Experimental} & \fbox{Community Intervention Trials} \\
& & & \fbox{Non-randomized CT} \\
& \fbox{Analytical} & & \\
& & & \fbox{Cohort} \\
\fbox{Study Design} & & & \fbox{Case-control} \\
& &\fbox{Observational} & \\
& & & \fbox{Cross-sectional} \\
& \fbox{Descriptive} & & \fbox{Other} \\
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\es
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\bs
\slidetitle{Randomized Clinical Trials (RCT)}
\begin{small}
\begin{itemize}
\item In their simplest implementation:
\begin{enumerate}
\item Subject enrolled into the study...
\item \underline{randomly assigned} to one of $\geq$ 2 treatments...
\item followed up until the outcome measure is obtained
\end{enumerate}
\item Using a prespecified effect measure, outcome comparisons made among various treatment groups
\item The key is that a random assignment determines subject's treatment
\item Strongest study design to establish causal relationships (superior control over confounding factors including those that are difficult or impossible to measure)
\item It may be beneficial to blind subjects and / or researchers to treatment assignment (avoid sources of bias)
\item Variations of this design include crossover studies
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Randomized Clinical Trials (RCTs)}
\begin{small}
\begin{itemize}
\item Situations favoring use of RCTs
\begin{itemize}
\item Exposure status is modifiable (subjects willing to relinquish control)
\item Legitimate uncertainty exists about effect of alternative interventions (e.g., the study is ethical)
\begin{itemize}
\item Reasonable to believe benefits of new treatment outweigh risks
\end{itemize}
\item Outcome of interest is reasonably common or effect of intervention on a rare outcome
is important enough to justify a large study
\end{itemize}
\item Factors to consider when recruiting / enrolling subjects
\begin{itemize}
\item Potential for benefit and/or risks of interventions
\item Internal validity: Subjects who will be reliable and compliant
\item External validity: Do results generalize to the broader population?
\item Power enhancement: high risk subjects
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Clinical Trial Examples}
\begin{itemize}
\item GUSTO 1: 41,021 patients in 1081 hospitals in 15 countries present with acute MI
\end{itemize}
\begin{footnotesize}
\psmatrix[colsep=1.8cm, rowsep=.3cm]
& \fbox{Treatment assigned} & \fbox{Percent Mortality} \\
& \fbox{Sk + subC Heparin} & $7.4 \%$ \\
& \fbox{Sk + IV Heparin}& $7.2 \%$\\
\fbox{Randomization} && \\
& \fbox{tPA + IV Heparin} & $6.3 \%$ \\
& \fbox{Sk + tPA + IV Heparin} & $7.0 \%$ \\
\endpsmatrix
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\begin{itemize}
\item P-value for tPA versus either Sk group 0.001
\item Significantly higher hemorrhagic strokes in tPA versus Sk
\end{itemize}
\end{footnotesize}
\es
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\bs
\slidetitle{Clinical Trials}
\begin{itemize}
\item Advantages
\begin{itemize}
\item Strong claims for causal effects
\item Optimal control over confounding factors
\end{itemize}
\item Challenges
\begin{itemize}
\item Very expensive
\item Time consuming
\item Ethical problems
\item Selection bias
\end{itemize}
\end{itemize}
\es
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\bs
\slidetitle{Cohort Studies (Follow-up Studies)}
\begin{small}
\begin{itemize}
\item In a classical cohort study
\begin{enumerate}
\item Study sample is identified...
\item exposure group status at the beginning of the study period is measured...
\item subjects are followed...
\item outcome status at the end of the study period is determined...
\end{enumerate}
\item Compare outcome status among various exposure groups.
\item Useful when clinical trials are infeasible
\begin{itemize}
\item For ethical reasons (e.g., when an exposure is thought to be harmful)
\item When exposure cannot be controlled (e.g., smoking, drug use, diet)
\end{itemize}
\item Frequently implemented when exposure is rare in the population
\begin{itemize}
\item By selecting a group of people who experience exposure variation we are able to detect exposure effects
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Cohort Studies}
\begin{small}
\begin{itemize}
\item Exposure not randomly assigned, only assessed
\begin{itemize}
\item Minimize confounding through study sample selection and through analysis
\end{itemize}
\item For this design to be useful, we need a sufficient number of subjects experiencing events
\item Prospective cohort studies
\begin{itemize}
\item Outcome events occur after study initiation
\end{itemize}
\item Retrospective cohort studies
\begin{itemize}
\item Outcomes have occurred by the time study is initiated
\item ``Reconstruct'' a prospective cohort study which has already occurred
\item Requires good exposure data on subjects at some time in the past
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Cohort Studies Example}
\begin{small}
Ray WA, Stein CM, Daugherty JR, Hall K, Arbogast PG, Griffin MR; {\it COX-2 selective non-steroidal and anti-inflammatory drugs and risk of serious coronary heart disease}, The Lancet, 2002.
\begin{itemize}
\item Study sample: 50-84 year olds participating in TennCare from January 1999 through June 2001 with no life-threatening non-cardiovascular disease
\begin{itemize}
\item $\approx$ 200,000 nonusers of NSAIDS
\item $\approx$ 24,000 Rofecoxib users
\item $\approx$ 150,000 other NSAIDs users
\end{itemize}
\begin{scriptsize}
\item Rate at which CHD occurred
\begin{center}
\begin{tabular}{lccc}
& per 1000 person-years & Adjusted Relative Rate \\
Non-users of NSAIDS & 13.0 & 1 \\
High dose Rofecoxib users & 21.0 & 1.70 (0.98-2.95; p=0.06)\\
New high dose Rofecoxib users & 24.0 & 1.93 (1.09-3.42; p=0.02)
\end{tabular}
\end{center}
\end{scriptsize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Cohort Studies}
\begin{itemize}
\item Advantages
\begin{itemize}
%\item No issues of temporality
\item Efficient for rare exposures
\item Can study numerous outcomes / responses
\item Useful when clinical trials are infeasible
\end{itemize}
\item Challenges
\begin{itemize}
\item No control over risk factor
\item Not efficient for rare outcomes
\item Potential for loss to followup
\item Expensive relative to other observational studies
\end{itemize}
\end{itemize}
\es
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\bs
\slidetitle{Case-Control Studies}
\begin{small}
\begin{itemize}
\item Subjects are identified based on disease (outcome) status and exposure status is assessed retrospectively
\begin{itemize}
\item Cases - those with disease
\item Controls - those without disease
\end{itemize}
\item In case-control studies we:
\begin{enumerate}
\item Identify cases...
\item Identify comparable controls...
\item Retrospectively determine prior exposure in cases and controls...
\end{enumerate}
\item Control identification
\begin{itemize}
\item Control group may be substantially different from cases
\item Must account for all differences between cases and controls that could explain the relationship between exposure and case status.
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Case-Control Studies: The Control Group}
\begin{small}
\begin{itemize}
\item {\blue Function of the control group:} to estimate the exposure that would have taken place in cases in the absence of an exposure-disease association.
\begin{itemize}
\item Can the exposure-disease relationship be attributed to other differences between cases and controls?
\end{itemize}
\item {\blue Sources of controls}
\begin{itemize}
\item Probability sample of non-cases from same population as cases
\begin{itemize}
\item {\blue Cases:} Females $\geq50$ years in TennCare with breast cancer (BC)
\item {\blue Controls:} Random sample of females $\geq50$ in TennCare without BC
\item Such controls are best but such a design may be difficult and expensive
\end{itemize}
\item Hospital or clinic-based
\begin{itemize}
\item {\blue Cases:} Females $\geq50$ years seeking treatment at Vanderbilt for BC
\item {\blue Controls:} Females $\geq50$ years seeking health care at Vanderbilt for non-BC related conditions
\item Challenge: May not be representative of study base (selection bias)
\end{itemize}
\item Relatives, friends or neighbors
\begin{itemize}
\item {\blue Controls:} Women living in the same block as BC cases, siblings of BC cases
\item Challenge: may be overmatched on genetic / demographic / occupational / behavioral / environmental predictors
\end{itemize}
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Case-Control Studies (cont.)}
\begin{small}
\begin{itemize}
\item Problems with exposure recall
\begin{itemize}
\item Cases and controls may not report exposure precisely (guessing exposure leads to issues with measurement error)
\item Recall bias: potentially problematic if cases recall exposure differently than controls
\begin{itemize}
\item Mother of a child with a birth defect may recall events that occurred during pregnancy that other mothers would not remember (e.g., respiratory infection)
\end{itemize}
\end{itemize}
\item CC studies are most useful when
\begin{itemize}
\item Disease is rare even among exposed subjects
\item Latency period from exposure to disease is very long
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Case-Control Studies}
\begin{small}
\begin{itemize}
\item Advantages
\begin{itemize}
\item Efficient for rare outcomes
\item Efficient for long latency periods
\item Can study numerous exposures
\item Optimal when cohort study is infeasible and RCT is unethical
\end{itemize}
\item Challenges
\begin{itemize}
\item Inefficient for rare exposures
\item Control identification
\item Confounding
\item Recall
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Study Design Summary}
\begin{small}
\begin{itemize}
\item Randomized Clinical Trial
\begin{itemize}
\item Optimal design for asserting a causal relationship between exposure and outcome
\end{itemize}
\item Cohort Study when...
\begin{itemize}
\item RCT is not possible or ethical
\item Exposure is rare
\end{itemize}
\item Case-Control Study when...
\begin{itemize}
\item Outcome is rare among all exposure groups
\item Biological latency period is long
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{\LARGE Frequency Measures}
\es
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\bs
\slidetitle{Incidence}
\begin{small}
\begin{itemize}
\item A measure of new events
\item {\blue Incidence Rate:}
\begin{itemize}
\item The rate of occurence of an outcome event
\item Number of events divided by person-time at risk of becoming diseased
\begin{align*}
\blue IR=\frac{\mbox{\# new events}}{\mbox{person-time at risk}}
\end{align*}
\item a.k.a., Incidence density
\item a.k.a., Hazard rate or force of morbidity (mortality) as person-time $\rightarrow 0$.
\item For rare diseases {\it IR} is usually multiplied by some constant
\begin{itemize}
\item In the UK the {\it IR} of prostate cancer is 74.3 per 100,000 men-years.
\end{itemize}
\end{itemize}
\end{itemize}
\end{small}
\es
\bs
\slidetitle{Incidence (cont.)}
\begin{small}
\begin{itemize}
\item {\blue Cumulative Incidence or Risk}
\begin{itemize}
\item Probability of a new event for individuals in a population over a specified time period
\begin{align*}
\blue CI=\frac{\mbox{\# new events in a population}}{\mbox{size of population at risk}}
\end{align*}
\item a.k.a., Incidence fraction
\item With $CI$ time period is implicit, as opposed to $IR$ where it is explicit
\begin{itemize}
\item e.g., the longer the observation time, the greater the risk
\end{itemize}
\end{itemize}
\item {\blue Note:} Constant IR implies non-constant increases in (additive) risk over time
\begin{center}
\begin{figure}
\includegraphics[height=4in, width=3.25in, angle=270]{IncidenceRates.ps}
\end{figure}
\end{center}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Prevalence}
\begin{small}
\begin{itemize}
\item A measure of status
\item Proportion of the population with disease at a given time point
\begin{align*}
\blue \mbox{Prevalence} & \blue = \ \frac{\mbox{\# existing cases}}{\mbox{population at risk of having disease}}
\end{align*}
\item Prevalence per 1000 = Prevalence $\times$ 1,000
\item {\blue Point prevalence:}
\begin{itemize}
\item Denominator is the population at risk at a given point in time
\end{itemize}
\item {\blue Period prevalence:}
\begin{itemize}
\item Numerator is the number of people who had disease at any point during the observation period
\item Denominator is the population at risk (usually) at the midpoint of the observation time
\end{itemize}
\item Effected by: Immigration, emigration, birth, death, and length of disease
\end{itemize}
\end{small}
\es
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\bs
\vspace*{-.4in}
\begin{center}
\begin{figure}
\includegraphics[height=3in, width=2.5in, angle=270]{incidence.ps}
\end{figure}
\end{center}
\begin{footnotesize}
\vspace*{-.55cm}
\begin{itemize}
\item Point prevalence at initiation:
\item Period prevalence:
\item CI:
\item IR: $\frac{ }{ \mbox{total person-time at risk of becoming diseased}}$
\end{itemize}
\end{footnotesize}
\es
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\bs
\slidetitle{\LARGE Measures of Association}
\es
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\bs
\slidetitle{Measures of Association}
\begin{small}
\begin{itemize}
\item {\blue Risk (R):} probability than an event will occur in an individual during the observation period
\begin{itemize}
\item We generally use the observed CI to estimate R
\begin{itemize}
\item CI is actually an estimate of average R
\end{itemize}
\item $R_{\overline{E}}$: risk for an individual in the unexposed group
\item $R_{E}$: risk for an individual in exposed group
\end{itemize}
\item {\blue Relative Risk (RR):} Ratio of the risk of an event among those who are exposed to E to the risk among those who are not exposed to E \\ \begin{center} {\red $RR = R_E / R_{\overline{E}}$.} \end{center}
%\begin{footnotesize}
\begin{itemize}
\item a.k.a., risk ratio
\item Multiplicative measure of excess risk (e.g. a ``fold increase")
\item {\blue Rate ratio} if interested in IR not CI (Vioxx study presented earlier)
\end{itemize}
%\end{footnotesize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Measures of Association}
\begin{small}
\begin{itemize}
\item {\blue Risk Difference, Absolute Risk Reduction, Attributable Risk (AR):} Difference in risk between exposed and unexposed subjects \\ \begin{center} {\red $AR= R_E - R_{\overline{E}}$} \end{center}
\begin{itemize}
\item A measure of the (absolute) reduction in risk that would occur in the exposed group if exposure is removed
\end{itemize}
\item {\blue Relative Risk Reduction, Attributable Risk Percent (AR$\%$):} Percentage of disease in exposed individuals caused by (??) exposure \\ \begin{center} {\red $AR$\%$=100 \times (R_E - R_{\overline{E}})/R_E$.}\end{center}
\begin{itemize}
\item A measure of $\%$ reduction in risk of disease for exposed subjects if exposure is removed
\end{itemize}
%\item {\blue Number Needed to Treat (NNT):} Number of persons treated to prevent one outcome or the reciprocal of AR {\red $NNT = 1/(R_1 - R_0)$}
\end{itemize}
\end{small}
\es
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\bs
{\blue Smoking and CHD: A Hypothetical Cohort Study of 3,000 Smokers and 5,000 Non-Smokers with 2 years of followup}
\begin{small}
\begin{center}
\begin{tabular}{lccc}
& Develop CHD & Do Not Develop CHD & Total \\
\hline Smoke & 84 & 2,916 & 3,000 \\
Do Not Smoke & 87 & 4,913 & 5,000 \\ \hline
\end{tabular}
\end{center}
\vspace*{.5cm}
\begin{itemize} \footnotesize
\item Risk among smokers: $ R_{S}=\frac{84}{3000}=0.028$ or 28 per 1000
\item Risk among non-smokers: $R_{NS}=\frac{87}{5000}=0.0174$ or 17.4 per 1000
\item Relative risk: $RR=\frac{0.028}{0.0174} \approx 1.61$
\item Attributable Risk: $AR=0.028-0.0174=0.0106$ or 10.6 per 1000
\item Attributable Risk Percent: $AR\%=100 \times (0.028-0.174)/0.028 \approx 38\%$
\end{itemize}
\end{small}
\es
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%\bs
%\slidetitle{Nausea in Ambulatory Surgery}
%\begin{itemize}
% \item Cohort study of the association between ambulatory surgery anethesia and peri-operative nausea
% \item Cohort: all patients undergoing gynecologic laparoscopy
% \item Predictor: Anathesia (general versus MAC)
% \item Outcome: Peri-operative nausea (yes/no).
% \item Results:
% \begin{itemize}
% \begin{small}
% \item 134 of 6030 using MAC result in nausea ($\approx 0.022$)
% \item 77 of 980 using general result in nausea ($\approx 0.$
% \item RR = $R_1 / R_0 = 0.022 / 0.079 = 0.28$ or $28\%$
% \item ARR = $R_0 - R_1 = 0.079 - 0.022 = 0.057$ or $5.7\%$
% \item RRR = $(0.079 - 0.022) / 0.079 =0.72$ or $72\%$
% \item NNT = $1/ARR = 1/0.057 \approx 18$
% \end{small}
% \end{itemize}
%\end{itemize}
%\es
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\bs
\slidetitle{Interpretations: Additive versus Multiplicative Effects}
\begin{small}
\begin{itemize}
\item $RR>1$ $\Leftrightarrow$ $AR>0$: Exposed subjects are more likely to experience the outcome than unexposed subjects
\item AR vs. RR
\begin{itemize}
\item AR is a measure of the additive excess risk due to exposure
\begin{itemize}
\item Easily translated directly into $ \$ $
\item Answers public health questions directly
\end{itemize}
\item RR is a measure of multiplicative excess risk due to exposure
\end{itemize}
\item Relationship between these measures and the relevance of each depends upon baseline risk (e.g., the risk among unexposed subjects)
\begin{align*}
AR = R_E - R_{\overline{E}} = RR \times R_{\overline{E}} - R_{\overline{E}} = R_{\overline{E}} \times (RR-1)
\end{align*}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Example: Impact of Baseline Risk on Measures of Excess Risk}
\begin{footnotesize}
\begin{tabular}{|l|c|c|c|} \hline
& Scenario 1 & Scenario 2 & Scenario 3 \\ \hline
$R_{\overline{E}}$ & $\frac{50}{1000}= 0.05$ & $\frac{200}{1000}=0.2$ & $\frac{2}{1000}=0.002$ \\
$R_E$ &$\frac{150}{1000}=0.15$ & $\frac{300}{1000}=0.3$ & $\frac{3}{1000}=0.003$\\
&&& \\
RR &${\blue \frac{0.15}{0.05} =3}$ & ${\blue \frac{0.3 }{ 0.2} = 1.5}$ & ${\blue \frac{0.003}{ 0.002 }= 1.5}$\\
AR$\%$ &${\blue 100\frac{(0.15-0.05)}{0.15}=67}$ & ${\blue 100\frac{(0.3-0.2)}{0.3} = 33}$ & ${\blue 100\frac{0.003 - 0.002 }{ 0.003 }= 33}$\\
&&& \\
AR &${\red 0.15-0.05 = 0.1}$ & $ {\red 0.3 - 0.2 = 0.1}$ & ${\red 0.003 - 0.002 = 0.001} $\\
NNT &${\red \frac{1}{0.1}=10}$ & $ {\red \frac{1}{0.1} = 10}$ & ${\red \frac{1 }{ 0.001} = 1000}$\\ \hline
\end{tabular}
\end{footnotesize}
\es
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\bs
\slidetitle{GUSTO 1}
\begin{small}
\begin{center}
\begin{tabular}{lccc}
Treatment Group & 30-day Mortality & AR & RR \\ \hline
Sk + SubC Hep & 7.4$\%$ & 0 & 1 \\
SK + IV Hep & 7.2$\%$ & -0.2 & 0.97 \\
tPA + IV Hep & 6.3$\%$ & -1.1 & 0.85 \\
tPA + Sk + IV Hep & 7.0$\%$ & -0.4 & 0.95 \\ \hline
\end{tabular}
\end{center}
Reference group is Sk + SubC Hep
\end{small}
\es
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\bs
\slidetitle{The Odds Ratio}
\begin{small}
\begin{itemize}
\item {\blue Odds:} The probability that an event occurs divided by the probability that it does not occur
\item Odds of disease for those exposed to E
\begin{align*}
Odds_{D|E}=\frac{P(D|E)}{P(\overline{D} | E)} \equiv \frac{p_E}{1-p_E}
\end{align*}
\item Odds of disease for those \textbf{NOT} exposed to E
\begin{align*}
Odds_{D|\overline{E}}=\frac{P(D|\overline{E})}{P(\overline{D} | \overline{E})} \equiv \frac{ p_{\overline{E}} }{ 1-p_{\overline{E}} }
\end{align*}
\item {\blue Odds ratio of disease from exposure E:} Odds of disease for those exposed to E divided by the odds of disease for those \textbf{NOT} exposed to E
\begin{align*}
OR_{D|E}=\frac{p_E}{1-p_E} \left/ \frac{p_{\overline{E}}}{1-p_{\overline{E}}} \right.
\end{align*}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{The Odds Ratio}
\begin{small}
\begin{center}
\begin{tabular}{l|cc|c}
\multicolumn{1}{c}{} & $D$ & \multicolumn{1}{c}{$\overline{D}$} & Total \\ \cline{2-3}
$E$ & A & B & A+B \\
$\overline{E}$ & C & D & C+D \\ \cline{2-3}
\multicolumn{1}{c}{Total}& A+C & \multicolumn{1}{c}{B+D} & A+B+C+D \\
\end{tabular}
\end{center}
\begin{align*}
P(D|E) = & \frac{A}{A+B} = \frac{P(D \mbox{ and } E)}{P(E)}
\end{align*}
\begin{itemize}
\item $Odds_{D|E} = \frac{A/(A+B)}{B/(A+B)} = \frac{A}{B}$
\item $Odds_{D|\overline{E}} = \frac{C/(C+D)}{D/(C+D)} = \frac{C}{D}$
\item $OR_{D|E} = \frac{A}{B} / \frac{C}{D} = \frac{A \cdot D}{B \cdot C}$
\item $OR_{E|D} = \frac{A}{C} / \frac{B}{D} = \frac{A \cdot D}{B \cdot C}$
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Key Mathematical Property of the Odds Ratio}
\begin{small}
\begin{itemize}
\item Reversibility
\item This property comes from the Bayes' Theorem:
\begin{small}
\begin{align*}
P(X|Y) = & \frac{P(Y \mbox{ and } X)}{P(Y)} = \frac{P(Y|X) \cdot P(X)}{P(Y)}
\end{align*}
\end{small}
\item Applied to the disease - exposure model:
\begin{small}
\begin{align*}
OR_{D|E} = \frac{ \frac{P(D|E)}{P(\overline{D}|E)} }{\frac{P(D|\overline{E})}{P(\overline{D}|\overline{E})} } = & \underbrace{ \frac{P(D|E)}{P(\overline{D}|E)} \cdot \frac{P(\overline{D}|\overline{E})}{P(D|\overline{E})}}_{\mbox{Apply Bayes now}} \\
= & \frac{P(E|D)}{P(\overline{E}|D)} \cdot \frac{P(\overline{E}|\overline{D})}{P(E|\overline{D})} = OR_{E|D}
\end{align*}
\end{small}
\item Makes evaluation of case-control studies possible
\item More transportable than RR (RR=2 can only apply if $p_{\overline{E}} \leq 0.5$)
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Odds Ratio and Case-Control Studies}
\begin{small}
\begin{itemize}
\item In CC studies, we collect information retrospectively but analyze it as if data were collected prospectively
\item Possible because logistic regression use to analyze case-control data
\begin{footnotesize}
\begin{align*}
I(Exposed) = & \begin{cases} 1 & \mbox{Exposed} \\ 0 & \mbox{Not Exposed} \end{cases} \\
log \left(\frac{p}{1-p}\right) = & \beta_0 + \beta_1 \cdot I(Exposed) + \mbox{confounders} \\
\Rightarrow \beta_1 = & log \left(\frac{p_E}{1-p_E}\right) - log \left(\frac{p_{\overline{E}}}{1-p_{\overline{E}}}\right)
= log \left[ \frac{p_E}{1-p_E} \left/ \frac{p_{\overline{E}}}{1-p_{\overline{E}}} \right. \right] \\
\Rightarrow e^{ \beta_1} = & \frac{p_E}{1-p_E} \left/ \frac{p_{\overline{E}}}{1-p_{\overline{E}}} \right.
\end{align*}
\end{footnotesize}
\item Since our responses are (log) odds ratios, we can make prospective inference with a retrospective study
\item What is $\beta_0$ in CC studies?
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{OR (RR) as an estimate of RR (OR) in Rare Diseases}
\begin{small}
\begin{center}
\begin{tabular}{lcc}
\multicolumn{1}{c}{} & Diseased & Total \\ \hline
Exposed & 50 & 10,000 \\
Not Exposed & 20 & 10,000 \\ \cline{2-3} \hline
\end{tabular}
\end{center}
\begin{itemize}
\item $R_E = \frac{50}{10,000} = 0.005$
\item $R_{\overline{E}} = \frac{20}{10,000} = 0.002 \Rightarrow RR = 2.5$
\item $Odds_{D|E} = \frac{50}{9,950} \approx 0.005$
\item $Odds_{D|\overline{E}} = \frac{20}{9,980} \approx 0.002$
$\Rightarrow OR \approx 2.5$
\item $OR \approx RR$
\item OR commonly called RR with rare diseases, but unlike RR, OR is valid for the entire risk spectrum
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Correlation $\rho$}
\begin{small}
\begin{itemize}
\item A measure of the \textbf{linear} association between two continuous random variables $X$ and $Y$
\begin{footnotesize}
\begin{align*}
\rho = \frac{E\left[(X-E(X)) \times (Y-E(Y))\right]}{\sqrt{Var(X)\times Var(Y)}}
\end{align*}
\end{footnotesize}
\item Estimated with the Pearson correlation coefficient
\begin{footnotesize}
\begin{align*}
r= \frac{ \sum_{i=1}^N(x_i-\overline{x})(y_i-\overline{y}) }{\sqrt{\sum_{i=1}^N (x_i-\overline{x})^2 \cdot \sum_{i=1}^N(y_i-\overline{y})^2}}
\end{align*}
\end{footnotesize}
\item $-1 \leq \rho \leq 1$
\item Simple linear regression model
\end{itemize}
\begin{align*}
E(Y | X) = \beta_0 + \beta_1 X = \beta_0 + \frac{\rho \sqrt{Var(Y)}}{\sqrt{Var(X)}} \times X
\end{align*}
\end{small}
\es
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\bs
\begin{center}
\begin{figure}
\includegraphics[height=4in, width=2.8in, angle=270]{Rhos.ps}
\end{figure}
\end{center}
\es
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\bs
\slidetitle{Analysis of Paired Observations}
\begin{small}
\begin{itemize}
\item In many types of experiments we measure multiple observations on an individual
\item Pre-post Analysis
\begin{itemize}
\item A baseline (pre-treatment) measurement is observed, treatment is administered, a followup (post-treatment) is observed
\end{itemize}
\item Cross-over Clinical Trial
\begin{itemize}
\item Subjects are randomized to a treatment order
\end{itemize}
\end{itemize}
\end{small}
\begin{footnotesize}
\psmatrix[colsep=1cm, rowsep=.2cm]
& \fbox{Order A} & \fbox{$Y_t$} & \fbox{Switch} &\fbox{ $Y_c$} \\
\fbox{Randomization} & & & \\
& \fbox{Order B} & \fbox{$Y_c$} & \fbox{Switch} & \fbox{$Y_t$} \\
\endpsmatrix
\ncline[linecolor=black,arrows=->]{2,1}{1,2}
\ncline[linecolor=black,arrows=->]{2,1}{3,2}
\ncline[linecolor=blue,arrows=->]{1,2}{1,3}\aput{:U}{\scriptsize Treated}
\ncline[linecolor=gray,arrows=->]{1,3}{1,4}\aput{:U}{\scriptsize Washout}
\ncline[linecolor=red,arrows=->]{1,4}{1,5}\aput{:U}{\scriptsize Control}
\ncline[linecolor=red,arrows=->]{3,2}{3,3}\aput{:U}{\scriptsize Control}
\ncline[linecolor=gray,arrows=->]{3,3}{3,4}\aput{:U}{\scriptsize Washout}
\ncline[linecolor=blue,arrows=->]{3,4}{3,5}\aput{:U}{\scriptsize Treated}
\end{footnotesize}
\es
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\bs
\slidetitle{Paired Data: Choosing an Effect Measure}
\begin{small}
\begin{itemize}
\item Additive vs. multiplicative effects
\item Key consideration: Effect of treatment should not depend on the baseline value
\item Objective method for choosing an effect measure with paired data
\item Plot $(Y_{2} + Y_{1})/2$ vs $Y_{2}-Y_{1}$
\begin{itemize}
\item If there is no trend between the average (or baseline) value and the difference, then the treatment effect does not depend on the baseline value and use the difference as an effect measure
\item If you see trends consider a transformation
\end{itemize}
\end{itemize}
\end{small}
\es
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\bs
\slidetitle{Paired Data: Scenario 1}
\vspace*{-.5cm}
\begin{center}
\begin{figure}
\includegraphics[height=3.5in, width=2.5in, angle=270]{additive.ps}
\end{figure}
\end{center}
\begin{footnotesize}
\begin{itemize}
\item Consider testing $H_0: median(Y_2-Y_1)=0$ using a Wilcoxon signed-rank test
\end{itemize}
\end{footnotesize}
\es
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\bs
\slidetitle{Paired Data: Scenario 2}
\vspace*{-.5cm}
\begin{center}
\begin{figure}
\includegraphics[height=4in, width=2.2in, angle=270]{multiplicative.ps}
\end{figure}
\end{center}
\begin{footnotesize}
\begin{itemize}
\item Consider a test such as $H_0: median\left[log(Y_2)-log(Y_1)\right]=0$ using a Wilcoxon signed-rank test
\item Note: $med\left(log(Y_2)-log(Y_1)\right) \equiv med\left[ log \left( \frac{Y_2}{Y_1}\right) \right] \equiv log\left[med\left(\frac{Y_2}{Y_1}\right)\right]$
\end{itemize}
\end{footnotesize}
\es
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\bs
\slidetitle{Review}
\begin{footnotesize}
\begin{itemize}
\item Study design
\begin{itemize}
\item Randomized clinical trials
\item Cohort studies
\item Case-control studies
\end{itemize}
\item Frequency measures
\begin{itemize}
\item Incidence
\item Prevalence
\end{itemize}
\item Measures of Association
\begin{itemize}
\item Attributable risk
\item Relative risk
\item Odds ratio
\item Correlation
\item Analysis of paired data (measure of change)
\end{itemize}
\end{itemize}
\end{footnotesize}
\es
\end{document}