Questions tagged [ratio]

the quantitative relation between two amounts showing the number of times one value contains or is contained within the other. Mathematically the quotient of one amount by another amount.

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45
votes
15answers
48k views

Expected number of ratio of girls vs boys birth

I have came across a question in job interview aptitude test for critical thinking. It is goes something like this: The Zorganian Republic has some very strange customs. Couples only wish to have ...
29
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3answers
2k views

I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?

The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
28
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4answers
26k views

How to compute the confidence interval of the ratio of two normal means

I want to derive the limits for the $100(1-\alpha)\%$ confidence interval for the ratio of two means. Suppose, $X_1 \sim N(\theta_1, \sigma^2)$ and $X_2 \sim N(\theta_2, \sigma^2)$ being independent, ...
16
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1answer
2k views

Ratios in Regression, aka Questions on Kronmal

Recently, randomly browsing questions triggered a memory of on off-hand comment from one of my professors a few years back warning about the usage of ratios in regression models. So I started reading ...
16
votes
1answer
571 views

An unbiased estimator of the ratio of two regression coefficients?

Suppose you fit a linear/logistic regression $g(y) = a_0 + a_1\cdot x_1 + a_2\cdot x_2$, with the aim of an unbiased estimate of $\frac{a_1}{a_2}$. You are very confident that both $a_1$ and $a_2$ ...
12
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3answers
16k views

What are the issues with using percentage outcome in linear regression?

I have a study where many outcomes are represented like percentages and I'm using multiple linear regressions to asses the effect of some categorical variables on these outcomes. I was wondering, ...
11
votes
2answers
3k views

Should types of data (nominal/ordinal/interval/ratio) really be considered types of variables?

So for instance here are the definitions that I get from standard text books Variable - characteristic of population or sample. ex. Price of a stock or grade on a test Data - actual ...
11
votes
3answers
2k views

What is the ratio of uniform and normal distribution?

Let $X$ follow a uniform distribution and $Y$ follow a normal distribution. What can be said about $\frac X Y$? Is there a distribution for it? I found the ratio of two normals with mean zero is ...
11
votes
1answer
4k views

Distribution of the ratio of dependent chi-square random variables

Assume that $ X = X_1 + X_2+\cdots+ X_n $ where $X_i \sim N(0,\sigma^2)$ are independent. My question is, what distribution does $$ Z = \frac{X^2}{X_1^2 + X_2^2 + \cdots + X_n^2}$$ follow? I know ...
11
votes
1answer
1k views

How to find the sample points that have statistically meaningful large outlier ratios between two values of the point?

As an example application, consider following two properties of Stack Overflow users: reputation and profile view counts. It is expected that for most users those two values will be proportional: high ...
10
votes
1answer
350 views

Expected value of maximum ratio of n iid normal variables

Suppose $X_1,...,X_n$ are iid from $N(\mu,\sigma^2)$ and let $X_{(i)}$ denote the $i$'th smallest element from $X_1,...,X_n$. How would one be able to upper bound the expected maximum of the ratio ...
10
votes
1answer
6k views

Testing Sharpe Ratio significance

What is the proper way to test the significance of Sharpe Ratios or Information Ratios? The Sharpe Ratios will be based on various equity indices and may have variable look-back periods. One solution ...
9
votes
1answer
1k views

If X/Y has the same distribution as Z, is it true that X has the same distribution as YZ?

Let X, Y and Z be three independent random variables. If X/Y has the same distribution as Z, is it true that X has the same distribution as YZ?
9
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1answer
1k views

Expected value of ratio of correlated random variables?

For independent random variables $\alpha$ and $\beta$, is there a closed form expression for $\mathbb E \left[ \frac{\alpha}{\sqrt{\alpha^2 + \beta^2}} \right]$ in terms of the expected values and ...
8
votes
3answers
838 views

How to correctly analyze fatality rate and daily deaths of Chinese and Italian COVID-19 outbreak?

This is a strange case of difference in fatality rate between Chinese and Italian covid-19 outbreak. In my knowledge, fatality rate is a ratio between deaths from a certain disease compared to the ...
8
votes
1answer
206 views

Do random variables follow the same algebraic rules as ordinary numbers?

In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim ...
8
votes
1answer
14k views

Is the ratio distribution of two normally distributed variables ever normal?

Let $Z = X / Y$ where $X$ and $Y$ are normally distributed random variables. Is $Z$ normally distributed for any $X$ and $Y$?
7
votes
3answers
961 views

If my goal is to test the absolute change of the ratios, can I compare the ratios directly without log transformation?

Ratios (e.g. $Z$=$Y$/$X$) are frequently used (e.g. fold-changes in mRNA or protein expression, body mass index [BMI], etc.). Many people advise that variables coded as ratios (e.g. fold-change) ...
7
votes
1answer
3k views

Using Fieller's theorem to calculate the confidence interval of a ratio (paired measurements)

If you have two means (with their own confidence intervals) and want to represent them as a ratio, how do calculate the confidence interval for the ratio? An answer that was given to me, mentions ...
6
votes
1answer
516 views

Ratio of correlated sample variances (gamma distributed)

for $N$ samples of two correlated random variables $X \sim N\left(0,\sigma_X^2\right)$ and $Y \sim N\left(0, \sigma_Y^2\right)$ with correlation $\rho$, I am analyzing the ratio of the sample ...
6
votes
1answer
703 views

How should I evaluate the expectation of the ratio of two random variables?

Let $A$ and $B$ be random variables and $f(A,B)=\frac{A}{B}$. How should I approximate $E(f(A,B))$? I think a Taylor expansion may be in order, but I am not sure how to fire it off in this function. ...
6
votes
1answer
1k views

Analysing ratios of variables

Ratios (X/Y, e.g. body mass index) are variables with odd distributions. They have no means or moments (e.g. variances, skewness or kurtosis). Thus, my questions are: How to compare between (or among)...
6
votes
1answer
331 views

Direct way of calculating $\mathbb{E} \left[ \frac{\textbf{h}^{H} \textbf{y}\textbf{y}^{H} \textbf{h}}{ \| \textbf{y} \|^{4} } \right]$

Considering the following random vectors: \begin{align} \textbf{h} &= [h_{1}, h_{2}, \ldots, h_{M}]^{T} \sim \mathcal{CN}\left(\textbf{0}_{M},d\textbf{I}_{M \times M}\right), \\[8pt] \textbf{w} &...
6
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0answers
190 views

Compare two set of ratios [closed]

I'm looking for a statistical test to compare two set of ratios. For example A = [(2,13), (10,50), (23, 86), (33, 133)] B = [(6,10), (21,33), (30,62), (44,80)] ...
6
votes
0answers
917 views

Why would I use ratio estimation instead of regression estimation to estimate means?

I am taking a graduate course on survey data analysis. I was recently introduced to ratio estimation and regression estimation. I understand that using ratio estimator may be easier if we are ...
5
votes
2answers
906 views

What is the distribution of the ratio between independent Beta and Gamma random variables?

What would be the distribution of the following equation: $$y = \frac{a}{(a+d)^2}$$ where $a, d$ $\sim$ $\Gamma(M,c)$. Additionally, $a$ and $d$ are independent random variables.
5
votes
3answers
773 views

median(a)/median(b) not equal median(a/b)

I'm sure this is a very straightforward question but it came up in my work today and I could not think of the reasoning behind it. I had two sets of numeric values (A & B) and was looking at the ...
5
votes
2answers
161 views

Distribution of the ratio of a Normal distribution divided by Lognormal distribution

I want to know the distribution (and the moments) of a variable, $Z = X/Y$, where $X\sim \mathcal{N}(\mu_{x}, \sigma^{2}_{x})$, and , $Y\sim \text{Lognormal}(\mu_{y},\sigma_{y})$? Hence, what I want ...
5
votes
1answer
109 views

How to find the expectation $\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right]$?

Given the independent and complex Gaussian random variables $h$ and $w$, how does one can find the following expectation? $$\mathbb{E} \left[ \frac{|h|^4}{|h+w|^2} \right] = \int_{\mathbb{C}}\...
5
votes
1answer
161 views

Can a ratio of random variables be normal? [duplicate]

For a pair of random variables $Y$ and $Z$, is it possible that their ratio $X:=\frac{Y}{Z}$ is (exactly, not asymptotically) normally distributed? If so, could you offer an example of the ...
5
votes
1answer
2k views

Beta regression and regression diagnostics. Do we need to check for normality and other diagnostics?

I have a dependent variable which is a ratio and 0 < y < 1 condition holds. I will apply betareg in Stata but I am not sure what are the diagnostics that are ...
5
votes
1answer
342 views

Disentangling the effect of a ratio from the effects of the numerator & denominator in linear models

My question: What is the most appropriate way to analyse cases in which two variables and their ratio are all believed to influence some outcome? I would ideally like guidance in a regression ...
5
votes
3answers
227 views

Generating identically-distributed random variables with a constraint

Is there a way to generate identically-distributed random variables (eg $x_1,x_2,x_3,x_4$) with the following constraint: $\frac{x_1*x_2}{x_3*x_4} ≡ 1$ $x \in (0,1)$ Please note that simply ...
5
votes
2answers
940 views

Uncertainty in a fractional count

What is the uncertainty (68% confidence level) of $N/M$, where $N$ is the number of entries that pass a cut and $M$ is the total number of entries? ($N$ and $M$ are both integers, and I'm interested ...
5
votes
1answer
183 views

Standard Error of Ratio of Weibull variates

Assuming that I have 2 distinct random variables that follow a Weibull distribution, what's the standard error* of the ratio of these two random variables? Basically I have $X \sim \text{Weibull}, Y \...
5
votes
3answers
182 views

What is the relationship between the ratio $y_{i}/x_{i}$ and the residuals of a linear regression $y_{i}$~$x_{i}$?

My background in statistical mathematics is poor. I don't know why $y_{i}/x_{i}$ and the residuals of the linear regression $y_{i}$~$x_{i}$ are always linearly related (whether X and Y are related or ...
5
votes
1answer
753 views

Distribution of the ratio between two Negative binomial distributions

I'm dealing with some biological data (read counts from gene expression measurements). For each individual we have 2 measurements, which correspond to two different version of the same gene. These ...
5
votes
2answers
231 views

What is the difference between controlling by dividing by a variable, and inserting it as a predictor?

I have a dataset, based on forum conversations, with the following variables: count of emotional words count of cognitive words ...
5
votes
1answer
990 views

Statistical test to assess differences between two sequencing methods

To asses differences between two DNA sequencing methods (454 and MiSeq) I have made artificial mixtures of different bacteria in the lab (mock communities). An Example might be (numbers are ...
5
votes
0answers
454 views

Distribution of ratio of Poisson distributed random variables

I am currently reading a paper and puzzling about a certain statement. I have a ratio $\frac{\hat\alpha_{T+1}}{\hat\alpha_{T}}=\frac{H_{T-4}+...+H_{T}}{H_{T-3}+...+H_{T+1}}\left(\frac{D_{T-3}+...+...
4
votes
3answers
1k views

Bias of estimating a Ratio?

I would like to know why 1/N * Summation of(y/x) is a worse estimator than average of y divided by average of x. This is in the context of constructing an estimator of a ratio where the ratio would ...
4
votes
1answer
191 views

What is the distribution of $x/(x + y)$ for $x$, $y$ independent and normally distributed with mean $0$

What is the distribution of $x/(x + y)$ for independent and normally distributed $x$, $y$ with mean $0$? I'm aware $x/y$ has a Cauchy distribution but I don't know if there is a way to make use of it. ...
4
votes
1answer
334 views

Ratio that accounts for different sample sizes

I'm trying to get a measure for a set of data that indicates someone's "spam score". Essentially, the higher the spam score, the more likely they are to be spammers. Right now, I'm measuring a person'...
4
votes
3answers
140 views

Distribution of *conditional* frequencies when frequencies follow a Dirichlet distribution

Context: we have a large number of individuals characterized by two binary traits; call these $T$ with values $\{0,1\}$, and $T'$ with values $\{0',1'\}$. So there are four types of individuals: $00'$,...
4
votes
1answer
198 views

One of the independent variables in a logistic regression is the ratio of other two inputs. Is this okay?

Suppose that I have a logistic regression with continuous independent variables $a$,$b$, and $c$. In my logistic regression, $c = \frac{a}{b}$. Is it all right to include variable $c$ in the ...
4
votes
1answer
246 views

Judging the quality of a statistical model for a percentage

I have a data set with multiple predictors, and a single response variable which is a percentage, thus is bounded between 0 and 100. I cannot share the dataset unfortunately. I would like to build a ...
4
votes
1answer
4k views

How to (properly) analyze the sex ratio

Few weeks ago, I was listening to a "HowToDoScience" lecture. In the section which focused on appropriate statistics in articles the lecturer said: Person who will analyse the sex ratio by the chi-...
4
votes
1answer
714 views

T-test or Z-test of two proportions?

We did an intervention to reduce the frequency of use of a certain type of treatment and now I want to compare the data pre vs post intervention. For each month I have the number of treatments per ...
4
votes
1answer
1k views

Calculating a risk ratio for specific x values from a GAM model using the mgcv package

I have fit a generalized additive model (GAM) using the mgcv package in R. My model has a dichotomous response variable and so i've used the binomial family link function. After creating the model I ...
4
votes
3answers
10k views

How to compare percentages for populations of different sizes?

I have several populations (of people, actually) which vary in size (from 5 to 6000). I would like to visualize the ratio of women vs. men in each of them so that they can be compared. I will get, for ...

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