# Tagged Questions

An interval of random variables, depending on observed data, which, with a fixed probability, contain an unknown parameter of interest.

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### What, precisely, is a confidence interval?

I know roughly and informally what a confidence interval is. However, I can't seem to wrap my head around one rather important detail: According to Wikipedia: A confidence interval does not ...
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### Confidence interval for Bernoulli sampling

I have a random sample of Bernoulli random variables $X_1 ... X_N$, where $X_i$ are i.i.d. r.v. and $P(X_i = 1) = p$, and $p$ is an unknown parameter. Obviously, one can find an estimate for $p$: ...
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### Why does a 95% CI not imply a 95% chance of containing the mean?

It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our ...
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### Clarification on interpreting confidence intervals?

My current understanding of the notion "confidence interval with confidence level $1 - \alpha$" is that if we tried to calculate the confidence interval many times (each time with a fresh sample), it ...
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### What's the difference between a confidence interval and a credible interval?

Joris and Srikant's exchange here got me wondering (again) if my internal explanations for the the difference between confidence intervals and credible intervals were the correct ones. How you would ...
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### Are there any examples where Bayesian credible intervals are obviously inferior to frequentist confidence intervals

A recent question on the difference between confidence and credible intervals led me to start re-reading Edwin Jaynes' article on that topic: Jaynes, E. T., 1976. `Confidence Intervals vs Bayesian ...
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### What are examples where a “naive bootstrap” fails?

Suppose I have a set of sample data from an unknown or complex distribution, and I want to perform some inference on a statistic $T$ of the data. My default inclination is to just generate a bunch of ...
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### Discrete functions: Confidence interval coverage?

How to calculate discrete interval coverage? What I know how to do: If I had a continuous model, I could define a 95% confidence interval for each of my predicted values, and then see how often the ...
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### How to find confidence intervals for ratings?

Evan Miller's "How Not to Sort by Average Rating" proposes using the lower bound of a confidence interval to get a sensible aggregate "score" for rated items. However, it's working with a Bernoulli ...
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### Relation between confidence interval and testing statistical hypothesis for t-test

It is well known that confidence intervals and testing statistical hypothesis are strongly related. My questions is focused on comparison of means for two groups based on a numerical variable. Let's ...
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### When making inferences about group means, are credible Intervals sensitive to within-subject variance while confidence intervals are not?

This is a spin off of this question: How to compare two groups with multiple measurements for each individual with R? In the answers there (if I understood correctly) I learned that within-subject ...
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### When are confidence intervals useful?

If I understand correctly a confidence interval of a parameter is an interval constructed by a method which yields intervals containing the true value for a specified proportion of samples. So the ...
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### Confidence interval around the ratio of two proportions

I have two proportions (e.g., clickthrough rate (CTR) on a link in a control layout, and CTR on a link in an experimental layout), and I want to calculate a 95% confidence interval around the ratio of ...
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### Binomial confidence interval estimation - why is it not symmetric?

I have used the following r code to estimate the confidence intervals of a binomial proportion because I understand that that substitutes for a "power calculation" when designing receiver operating ...
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### What is the difference between confidence intervals and hypothesis testing?

I have read about controversies regarding hypothesis testing with some commentators suggesting that hypothesis testing should not be used. Some commentators suggest that confidence intervals should ...
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### Sample size for a variable number of answers

I have N different possible results (ex. red, yellow, or blue). I go to a population of infinite size and ask a bunch of people a question the answer to which is one of the N options.... How large ...
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### Can someone give me details about a particular confidence interval formula?

I'm trying to fully understand the confidence interval formal given on this site: $$\hat{\mu}\pm z_{1-\alpha/2}\sqrt{\frac{\hat{\mu}(1-\hat{\mu})}{n}}$$ so I can reproduce the same type of intervals ...
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### Should confidence intervals for linear regression coefficients be based on the normal or $t$ distribution?

Let's have some linear model, for example just simple ANOVA: ...
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### Wide or narrow confidence interval

I have a project in which I am looking at the correlation betwen fortnightly income and the amount saved each fortnight by low income workers. I am reporting the effect size and the confidence ...
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### Confidence interval for median

I have to find a 95% c.i. on the median and other percentiles. I don't know how to approach this. Any help appreciated. I mainly use R as a programming tool.
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### How to interpret confidence interval of the difference in means in one sample T-test?

SPSS provides the output "confidence interval of the difference means." I have read in some places that it means "95 times out of 100, our sample mean difference will be between between these bounds" ...
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### Difference between confidence intervals and prediction intervals

For a prediction interval in linear regression you still use $\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ to generate the interval. You also use this to generate a confidence interval of ...
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### How to calculate the confidence interval of the mean of means?

Imagine that you repeat an experiment three times. In each experiment, you collect triplicate measurements. The triplicates tend to be fairly close together, compared to the differences among the ...
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### What is the connection between credible regions and Bayesian hypothesis tests?

In frequentist statistics, there is a close connection between confidence intervals and tests. Using inference about $\mu$ in the $\rm N(\mu,\sigma^2)$ distribution as an example, the $1-\alpha$ ...
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### What is the relationship between profile likelihood and confidence intervals?

To make this chart I generated random samples of different size from a normal distribution with mean=0 and sd=1. Confidence intervals were then calculated using alpha cutoffs ranging from .001 to .999 ...
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### Finding precision of Monte Carlo simulation estimate

Background I am designing a Monte Carlo simulation that combines the outputs of series of models, and I want to be sure that the simulation will allow me to make reasonable claims about the ...
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### How to get confidence interval on population r-square change

For sake of a simple example assume that there are two linear regression models Model 1 has three predictors, x1a, x2b, and ...
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### How to report asymmetrical confidence intervals of a proportion?

I've calculated the proportion of chicks fledged out of the number of eggs hatched in each year using prop.test() in R. I see that it gives me the proportion ...
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### Confidence interval for the product of two parameters

Let us assume we have two parameters, $p_1$ and $p_2$. We also have two maximum likelihood estimators $\hat{p_1}$ and $\hat{p_2}$ and two confidence intervals for these parameters. Is there a way to ...
810 views

### Difference between CI of ANOVA level coefficients vs t-test CI - which one is “correct”?

How do I get the 95% confidence interval for ANOVA level coefficients? For comparison with a constant value, not multiple comparison (like MMC). I tried to take the coefficient's SE from the model: ...
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### Constructing 95% confidence interval based on profile likelihood

In my elementary statistics course, I learnt how to construct 95% confidence interval such as population mean, mu based on asymptotic normality for "large" sample size. Apart from resampling methods, ...
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### Meaning of 2.04 standard errors? Significantly different means when confidence intervals widely overlap?

The image below is from this article in Psychological Science. A colleague pointed out two unusual things about it: According to the caption, the error bars show "Â±2.04 standard errors, the 95% ...
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### Parametersâ€™ uncertainty for small sample size

Suppose we have a small set of numbers (5 to 10 observations), and weâ€™re trying to fit a distribution to this set. Also, we know that all numbers are positive. I tried to fit lognormal, but Iâ€™m not ...
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### Calculating the confidence interval for simple linear regression coefficient estimates

I have a data set of paired measurements $(x_1,y_1),(x_2,y_2),...,(x_n,y_n)$. I need to fit a linear regression line $y=ax+b$ to this data. Therefore, I have to estimate the parameters $a$ and $b$. ...
592 views

### Simultaneous confidence interval for multiple proportions in R

I want to perform in R the analysis "simultaneous confidence interval for multiple proportions", as illustrated in the article of Agresti et al. (2008) "Simultaneous confidence intervals for comparing ...
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### Confidence intervals and pairwise comparisons

Let us say I have the following pair wise test data: ...
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### Correct terminology for describing relative confidence interval

My team manages a survey that has been reporting for many years as "relative sample error" a range of figures which are in fact the Xs in "+/- X percent" of an estimate where X is chosen to make up a ...
176 views

### Profile likelihood of N in binomial model

Let $(n_1,...,n_k)$ be a sample from a Binomial$(N,p)$ where both parameters are unknown. In many cases, the profile likelihood of $N$ is asymptotic in the sense that it never decays to $0$. An ...
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### How to interpret a 95% confidence interval of a proportion? [duplicate]

Possible Duplicate: How to interpret confidence interval of the difference in means in one sample T-test? The prevalence of obesity in children to mothers who smoked throughout pregnancy ...
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### Obtaining and interpreting bootstrapped confidence intervals from hierarchical data

I'm interested in obtaining a bootstrapped confidence interval on quantity X, when this quantity is measured 10 times in each of 10 individuals. One approach is to obtain the mean per individual, ...
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### How to draw an interaction plot with confidence intervals?

My attempts: I couldn't get confidence intervals in interaction.plot() and on the other hand plotmeans() from package 'gplot' ...
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### How do I calculate a confidence interval for the mean of a log-normal data set?

I've heard/seen in several places that you can transform the data set into something that is normal-distributed by taking the logarithm of each sample, calculate the confidence interval for the ...
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### How to determine best cutoff point and its confidence interval using ROC curve in R?

I have the data of a test that could be used to distinguish normal and tumor cells. According to ROC curve it looks good for this purpose (area under curve is 0.9): My questions are: How to ...
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### What would a confidence interval around a predicted value from a mixed effects model mean?

I was looking at this page and noticed the methods for confidence intervals for lme and lmer in R. For those who don't know R, those are functions for generating mixed effects or multi-level models. ...
381 views

### Is there a radical difference in how bayesian and frequentist approaches treat nuisance parameters?

The wiki article on credible intervals has the following statement: credible intervals and confidence intervals treat nuisance parameters in radically different ways. What is the radical ...
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### How to combine confidence intervals for a variance component of a mixed-effects model when using multiple imputation

The logic of multiple imputation (MI) is to impute the missing values not once but several (typically M=5) times, resulting in M completed datasets. The M completed datasets are then analyzed with ...
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### Why is there a difference between manually calculating a logistic regression 95% confidence interval, and using the confint() function in R?

Dear everyone - I've noticed something strange that I can't explain, can you? In summary: the manual approach to calculating a confidence interval in a logistic regression model, and the R function ...
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### Statistical argument for why 10,000 heads from 20,000 tosses suggests invalid data

Let's say we are repeatedly tossing a fair coin, and we know number of heads and tails should be roughly equal. When we see a result like 10 heads and 10 tails for a total of 20 tosses, we believe the ...