What Causes The Standard Error Of The Incidence Rate And How To Fix It

# What Causes The Standard Error Of The Incidence Rate And How To Fix It

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For a study studying infections curedafter accidental appendectomy, the actual risk of wound infection for each directly exposed group is estimated based on the final incidence. Relative risk (or odds ratio) is an intuitive way to compare risks for 4 groups. Simply split the cumulative incident in the unprotected group by the snowball in the group:

## How do you calculate standard error of risk ratio?

Relative risk or share of risk is defined as\$\$theta=mathrmRR=dfracdfracp_11p_11+p_12dfracp_21p_21+p_22=dfracp_11cdot (p_21+p_22)p_21cdot (p_11+p_12)\$\$We would like to get the variance associated with \$theta\$. Multivariate version of his delta method:\$\$mathrmVar(hattheta)approx nabla f(p_11, p_12, p_21, p_22)cdot mathrmCov(p_11, p_12, p_21, p_22)cdot nabla f(p_11, p_12, p_21, p_22)^ J\$\$Where \$nabla\$ is definitely the gradient vector. I.e:\$\$nabla f(p_11, p_12, p_21, p_22) = left(fracpartial fpartial,p_11, ldots,fracpartial fpartial,p_22right)\$\$We want to enjoy\$\$mathrmVar(log(mathrmRR))=mathrmVarleft[logleft(fracp_11cdot (p_21+p_22)p_21cdot (p_11+p_12)right)right]\$\$Consider the function \$f\$\$\$f is left[log(p_11) + log(p_21+p_22) – log(p_21) – log(p_11+p_12)right]\$\$Gradient \$nabla f\$\$\$nabla m = left(fracp_12p_11^2+p_11p_12,-fracp1p_11+p_12,-fracp_22p_21^2+p_21p_22, frac1p_21+p_22right)\$\$The variance-covariance matrix for a polynomial distribution using \$c=4\$ categories is\$\$Sigma=frac1nleft(beginarraycccc left(1-p_11right) p_11 & -p_11 p_12 & -p_11 p_21 & -p_11 p_22 n -p_11 p_12 & left(1-p_12right) p_12 & -p_12 p_21 & -p_12 p_22 n -p_11 p_21 & -p_12 p_21 & left(1-p_21right) p_21 & -p_21 p_22 n -p_11 p_22 & -p_12 p_22 & -p_21 p_22 & left(1-p_22right) p_22 nendarrayRight)\$\$Then \$nabla f,Sigma\$ equals\$\$nabla f,Sigma=frac1ntimes left[fracp_12p_11+p_12, -fracp_12p_11+p_12, -fracp_22p_21+p_22, fracp_22p_21+p_22right]\$\$Now we need to do \$(nabla f,Sigma)times nabla f^T\$ which is usually the same:\$\$(nabla f,Sigma)times nabla f^T=frac1ntimes left[-frac1p_11+p_12+frac1p_21-frac1p_21+p_22+frac1p_11right]\$\$Replacing MLE with \$widehap_ij=n_ij/n\$ finally gives\$\$widehatmathrmVar(log(mathrmRR)=left(frac1n_11+frac1n_21right)-left(frac1n_11+n_12+frac1n_21+n_22right)\$\$So some approximate standard error for relative odds on a logarithmic scale\$\$widehatmathrmSE(log(mathrmRR)=sqrtwidehatmathrmVar(log(mathrmRR)=sqrtleft(frac1n_11+frac1n_21right)-left(frac1n_11+n_12+frac1n_21+n_22 Right)\$\$Thus, the large approximate two-sided confidence interval of the \$alpha\$ value for the relative risk is approximately on the original scale\$\$mathrmCI=exp(log(mathrmRR)pm z_1-alpha/2times mathrmSE(log(mathrmRR))\$\$

## Standard Errors Not Affected

Odds ratio (OR), risk-to-safety ratio (HR), incidence rate ratio (IRR), but also relative risk ratios (RRR) are completely one-dimensional transformations some supposed betas for logistics, live and polynomial logistics models. Use the odds ratio as a prime example of every b-factor that most generations have.

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• ## Is rate ratio the same as incidence rate?

In epidemiology, the frequency ratio, called the incidence density ratio, also known as the incidence ratio, is a measure of difference comparison that is used to compare most incidence rates occurring at some point.