14
votes
Why would someone plot variance normalized by the mean?
Variance over mean is known as the Index of dispersion.
This can be useful when comparing two random variables with different means, in order to account for larger variance for larges means.
Example: ...
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12
votes
Accepted
How to measure dispersion in word frequency data?
For probabilities (proportions or shares) $p_i$ summing to 1, the family $\sum p_i^a [\ln (1/p_i)]^b$ encapsulates several proposals for measures (indexes, coefficients, whatever) in this territory. ...
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10
votes
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How to correct underdispersion in logistic regression
Getting a residual mean deviance around 0.63 is perfectly normal for binary regression and it does not indicate underdispersion or overdispersion. For binary regression, the residual deviance is ...
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10
votes
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Tweedie Dispersion Parameter Estimation Methods
Tweedie generalized linear models assume a mean-variance relationship with variance power $p$, defined by
$$E(y_i)=\mu_i$$
and
$${\rm var}(y_i)=\phi \mu_i^p$$
where $y_i$ is the $i$th observation, $\...
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10
votes
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Unusual variance measure
It's definitely not the most popular. You would not find it discussed among the commonly used statistics used for measuring dispersion.
The closest thing to it, that is quite popular is range, which ...
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9
votes
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Definition of exponential family with dispersion parameter
The definition you quote which is used with generalized linear models (glm) is not an exponential family, it is an exponential dispersion family. For a fixed value of the dispersion parameter $\phi$ ...
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9
votes
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Median of the squared difference from the median of a Cauchy random variable
Suppose $F$ is the distribution function of a random variable $X$ with median $m.$
By definition, a median is any number for which $F(m)\ge 1/2$ and $F(x) \le 1/2$ for all $x \lt m.$
For any non-...
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8
votes
Is mean deviation the same as mean absolute difference?
Mean deviation is the same as mean absolute deviation; it is mean deviation from the mean.
$$
MAD=\frac{1}{N}\sum_{i=1}^{N}|x_i-\overline{x}|
$$
Mean absolute difference is for two independent ...
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8
votes
How to measure dispersion in word frequency data?
I don't know if there's a common way of doing it, but this looks to me analogous to inequality questions in economics. If you treat each word as an individual and their count as comparable to income, ...
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8
votes
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How does R find the dispersion parameter in a GLM?
You can look in the code for summary.glm where you'll see:
...
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8
votes
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dispersion parameter in Poisson models
That's correct!
You've found out why glm doesn't use deviance/df as an estimate of dispersion: it's not a very good one. It uses the better estimate based on the ...
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7
votes
Accepted
GLM Tweedie dispersion parameter
For the benefit of other readers, Tweedie glms assume that the variance of the responses has the form
$$
{\rm var}(y_i) = \phi \mu_i^\xi
$$
where $\phi$ is the dispersion and $\xi$ is the variance ...
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7
votes
Concept of a z-score for a gamma distribution
The $z$ score expressed how many standard deviations a given observation from a symmetric distribution is away from the mean. Negative $z$ scores indicate an observation below, positive $z$ scores ...
- 131k
7
votes
Accepted
What do we mean by 'range' in descriptive statistics?
In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed.
(Wikipedia)
It does not say “from the central value”. ...
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6
votes
How to measure dispersion in word frequency data?
This article has a review of standard dispersion measures used by linguists. They are listed as single-word dispersion measures (They measure the dispersion of words across sections, pages etc.) but ...
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6
votes
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Variance (maybe?) of categorical data
You can always look at the variance of the counts, but looking at your description, entropy seems to be a natural choice, since it meets all of your criteria. Entropy is defined as
$$
S = -\sum_i p_i ...
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6
votes
How to compare the degree of dispersion of different distributions?
Comment continued:
Suppose you have $n = 100$ observations x
with summary statistics and stripchart as follows:
...
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6
votes
Accepted
Why does log-linear analysis seem to ignore the Poisson regression equidispersion assumption?
Quoting from Section 4.3.3 of the second edition of Agresti's "Categorical Data Analysis":
Overdispersion is common in the modeling of counts. When the model for the mean is correct but the ...
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5
votes
Accepted
Can variation ratio ever be 0?
According to Wikipedia, the variation ratio is defined as
$v := 1 - \frac{f_m}{N}$
where $f_m$ is the frequency of the mode and N is the number of observations.
Imagine that there is 1 observation, ...
- 1,207
5
votes
Why don't dispersions like median deviation and mode deviation exists on the lines of mean deviation?
Yes we can, there is, for example, median absolute deviation (MAD):
$$ \operatorname{MAD}(X) = \operatorname{median}\left(\ \left| X_{i} - \operatorname{median} (X) \right|\ \right)$$
It has even ...
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5
votes
What is the dispersion parameter of binomial distribution?
Using the definition of an exponential dispersion family at Definition of exponential family with dispersion parameter, we find that for the binomial family it is $a(\phi)=1$. Copying that definition ...
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5
votes
What do we mean by 'range' in descriptive statistics?
The range is twice the maximum absolute deviation from the mid-range, a tenable 'central value'; so on the face of it ought to pass muster as a measure of dispersion around that.
Formal criteria that ...
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4
votes
How does dispersion parameter affects results of gamma glm?
Short answer (I hope to come back and expand on this): dispersion parameter definitely affects likelihood, deviance, confidence intervals, p-values, pseudo-R^2. (Not coefficients. Log-likelihood, ...
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4
votes
How to measure dispersion in word frequency data?
The first I would do is calculating Shannon's entropy. You can use the R package infotheo, function entropy(X, method="emp"). If ...
- 3,049
4
votes
Accepted
Vector-valued estimators, intuitively why $var(\widetilde{\beta})-var(\widehat{\beta})$ being p.s.d. means $\widehat{\beta}$ more efficient?
Think about a vector $a$ specifying a linear combination of the $\hat\beta$s -- a direction in $\hat\beta$ space.
One natural way to extend the one-dimensional comparison based on variances is to say ...
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4
votes
Significant dispersion test
I'm the developer of DHARMa. This is a pretty common question, and there is a section in the vignette giving guidance on that, but I just realised now that it's not very well placed, so I will change ...
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4
votes
Accepted
Reference for Directional Statistics of Plane Orientation
On spheres (embedded in Euclidean space), a natural and useful metric for comparing two points is based on the cosine of their included angle,
$$\delta(z_1,z_2) = 1 - z_1\cdot z_2$$
where $|z_1| = |...
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3
votes
Dispersion value with glmmTMB versus mgcv::gam()
You can use the function sigma() to get the dispersion parameter (on the real rather than log scale). To interpret the dispersion parameters of any distribution, see ?sigma.glmmTMB.
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3
votes
How to measure dispersion in word frequency data?
One possible measure of equality you could use is the scaled Shannon entropy. If you have a vector of proportions $\boldsymbol{p} \equiv (p_1, ... , p_n)$ then this measure is given by:
$$\bar{H}(\...
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3
votes
Which measure of dispersion is this function related to?
To facilitate analysis, define the $c$-raw-moment estimator:
$$m_c \equiv \frac{1}{n} \sum_{i=1}^n x_i^c.$$
With a little algebra, your measure can be rewritten as:
$$r_c =\frac{m_c}{m_1^c} $$
...
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