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1. Making some assumptions, the prevalence of a disease in a population at a particular time can be obtained by sampling simple random sampling of people and finding who is sick. That is a binomial random variable and the Wald confidence interval for a proportion $p$ is

$$ p \pm 1.96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}$$

The variance portion is bounded above by 0.5, so we can make the simplifying assumption that the width of the confidence interval is $\sim 2/\sqrt{n}$. So, the answer to this part is that the confidence interval for $p$ decreases like $1/\sqrt{n}$. Quadruple your sample, halve your interval. Now, this was based on using a Wald interval, which is known to be problematic when $p$ is near 0 or 1, but the spirit remains the same for other intervals.

2. You need to look at metrics like specificity and sensitivity.

Sensitivity is the probability that a diseased person will be identified as diseased (i.e. tests positive). Specificity is the probability that a person without the disease is identified as not having the disease (i.e. tests negative). There are lots of other metrics for diagnostic tests found here which should answer your question.

3. I guess this is still up in the air. There are several attempts to model the infection over time. SIR models and their variants can make a simplifying assumption that the population is closed (i.e. S(t) + I(t) + R(t) = 1) and then I(t) can be interpreted as the prevalence. This isn't a very good assumption IMO because clearly the population is not closed (people die from the disease). As for modelling the diagnostic properties of a test, those are also a function of the prevalence. From Bayes rule

$$ p(T+ \vert D+) = \dfrac{P(D+\vert T+)p(T+)}{p(D+)}$$

Here, $P(D+)$ is the prevalence of the disease, so as this changes then the sensitivity should change as well.

1. Making some assumptions about the population size (namely that it is large enough that a binomial model is appropriate), the prevalence of a disease in a population at a particular time can be obtained by sampling simple random sampling of people and finding who is sick. That is a binomial random variable and the Wald confidence interval for a proportion $p$ is

$$ p \pm 1.96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}$$

The variance portion is bounded above by 0.5, so we can make the simplifying assumption that the width of the confidence interval is $\sim 2/\sqrt{n}$. So, the answer to this part is that the confidence interval for $p$ decreases like $1/\sqrt{n}$. Quadruple your sample, halve your interval. Now, this was based on using a Wald interval, which is known to be problematic when $p$ is near 0 or 1, but the spirit remains the same for other intervals.

2. You need to look at metrics like specificity and sensitivity.

Sensitivity is the probability that a diseased person will be identified as diseased (i.e. tests positive). Specificity is the probability that a person without the disease is identified as not having the disease (i.e. tests negative). There are lots of other metrics for diagnostic tests found here which should answer your question.

3. I guess this is still up in the air. There are several attempts to model the infection over time. SIR models and their variants can make a simplifying assumption that the population is closed (i.e. S(t) + I(t) + R(t) = 1) and then I(t) can be interpreted as the prevalence. This isn't a very good assumption IMO because clearly the population is not closed (people die from the disease). As for modelling the diagnostic properties of a test, those are also a function of the prevalence. From Bayes rule

$$ p(T+ \vert D+) = \dfrac{P(D+\vert T+)p(T+)}{p(D+)}$$

Here, $P(D+)$ is the prevalence of the disease, so as this changes then the sensitivity should change as well.

1. Making some assumptions, the prevalence of a disease in a population at a particular time can be obtained by sampling simple random sampling of people and finding who is sick. That is a binomial random variable and the Wald confidence interval for a proportion $p$ is

$$ p \pm 1.96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}$$

The variance portion is bounded above by 0.5, so we can make the simplifying assumption that the width of the confidence interval is $\sim 2/\sqrt{n}$. So, the answer to this part is that the confidence interval for $p$ decreases like $1/\sqrt{n}$. Quadruple your sample, halve your interval. Now, this was based on using a Wald interval, which is known to be problematic when $p$ is near 0 or 1, but the spirit remains the same for other intervals.

2. You need to look at metrics like specificity and sensitivity.

Sensitivity is the probability you test positive for the disease given you actually have the disease. Specificity is the probability you test negative for the disease given you are healthy. There are lots of other metrics for diagnostic tests found here which should answer your question.

3. I guess this is still up in the air. There are several attempts to model the infection over time. SIR models and their variants can make a simplifying assumption that the population is closed (i.e. S(t) + I(t) + R(t) = 1) and then I(t) can be interpreted as the prevalence. This isn't a very good assumption IMO because clearly the population is not closed (people die from the disease). As for modelling the diagnostic properties of a test, those are also a function of the prevalence. From Bayes rule

$$ p(T+ \vert D+) = \dfrac{P(D+\vert T+)p(T+)}{p(D+)}$$

Here, $P(D+)$ is the prevalence of the disease, so as this changes then the sensitivity should change as well.

1. Making some assumptions, the prevalence of a disease in a population at a particular time can be obtained by sampling simple random sampling of people and finding who is sick. That is a binomial random variable and the Wald confidence interval for a proportion $p$ is

$$ p \pm 1.96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}$$

The variance portion is bounded above by 0.5, so we can make the simplifying assumption that the width of the confidence interval is $\sim 2/\sqrt{n}$. So, the answer to this part is that the confidence interval for $p$ decreases like $1/\sqrt{n}$. Quadruple your sample, halve your interval. Now, this was based on using a Wald interval, which is known to be problematic when $p$ is near 0 or 1, but the spirit remains the same for other intervals.

2. You need to look at metrics like specificity and sensitivity.

Sensitivity is the probability that a diseased person will be identified as diseased (i.e. tests positive). Specificity is the probability that a person without the disease is identified as not having the disease (i.e. tests negative). There are lots of other metrics for diagnostic tests found here which should answer your question.

3. I guess this is still up in the air. There are several attempts to model the infection over time. SIR models and their variants can make a simplifying assumption that the population is closed (i.e. S(t) + I(t) + R(t) = 1) and then I(t) can be interpreted as the prevalence. This isn't a very good assumption IMO because clearly the population is not closed (people die from the disease). As for modelling the diagnostic properties of a test, those are also a function of the prevalence. From Bayes rule

$$ p(T+ \vert D+) = \dfrac{P(D+\vert T+)p(T+)}{p(D+)}$$

Here, $P(D+)$ is the prevalence of the disease, so as this changes then the sensitivity should change as well.

1. Making some assumptions, the prevalence of a disease in a population at a particular time can be obtained by sampling people and finding who is sick. That is a binomial random variable and the Wald confidence interval for a proportion $p$ is

$$ p \pm 1.96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}$$

The variance portion is bounded above by 0.5, so we can make the simplifying assumption that the width of the confidence interval is $\sim 2/\sqrt{n}$. So, the answer to this part is that the confidence interval for $p$ decreases like $1/\sqrt{n}$. Quadruple your sample, halve your interval. Now, this was based on using a Wald interval, which is known to be problematic when $p$ is near 0 or 1, but the spirit remains the same for other intervals.

2. You need to look at metrics like specificity and sensitivity.

Sensitivity is the probability you test positive for the disease given you actually have the disease. Specificity is the probability you test negative for the disease given you are healthy. There are lots of other metrics for diagnostic tests found here which should answer your question.

3. I guess this is still up in the air. There are several attempts to model the infection over time. SIR models and their variants can make a simplifying assumption that the population is closed (i.e. S(t) + I(t) + R(t) = 1) and then I(t) can be interpreted as the prevalence. This isn't a very good assumption IMO because clearly the population is not closed (people die from the disease). As for modelling the diagnostic properties of a test, those are also a function of the prevalence. From Bayes rule

$$ p(T+ \vert D+) = \dfrac{P(D+\vert T+)p(T+)}{p(D+)}$$

Here, $P(D+)$ is the prevalence of the disease, so as this changes then the sensitivity should change as well.

1. Making some assumptions, the prevalence of a disease in a population at a particular time can be obtained by sampling simple random sampling of people and finding who is sick. That is a binomial random variable and the Wald confidence interval for a proportion $p$ is

$$ p \pm 1.96\dfrac{\sqrt{p(1-p)}}{\sqrt{n}}$$

The variance portion is bounded above by 0.5, so we can make the simplifying assumption that the width of the confidence interval is $\sim 2/\sqrt{n}$. So, the answer to this part is that the confidence interval for $p$ decreases like $1/\sqrt{n}$. Quadruple your sample, halve your interval. Now, this was based on using a Wald interval, which is known to be problematic when $p$ is near 0 or 1, but the spirit remains the same for other intervals.

2. You need to look at metrics like specificity and sensitivity.

Sensitivity is the probability you test positive for the disease given you actually have the disease. Specificity is the probability you test negative for the disease given you are healthy. There are lots of other metrics for diagnostic tests found here which should answer your question.

3. I guess this is still up in the air. There are several attempts to model the infection over time. SIR models and their variants can make a simplifying assumption that the population is closed (i.e. S(t) + I(t) + R(t) = 1) and then I(t) can be interpreted as the prevalence. This isn't a very good assumption IMO because clearly the population is not closed (people die from the disease). As for modelling the diagnostic properties of a test, those are also a function of the prevalence. From Bayes rule

$$ p(T+ \vert D+) = \dfrac{P(D+\vert T+)p(T+)}{p(D+)}$$

Here, $P(D+)$ is the prevalence of the disease, so as this changes then the sensitivity should change as well.