26

I prefer to do power analyses beyond the basics by simulation. With precanned packages, I am never quite sure what assumptions are being made. Simulating for power is quite straight forward (and affordable) using R. decide what you think your data should look like and how you will analyze it write a function or set of expressions that will simulate the ...


25

You're on the right track, but always have a look at the documentation of the software you're using to see what model is actually fit. Assume a situation with a categorical dependent variable $Y$ with ordered categories $1, \ldots, g, \ldots, k$ and predictors $X_{1}, \ldots, X_{j}, \ldots, X_{p}$. "In the wild", you can encounter three equivalent choices ...


22

To manually verify the predictions derived from using polr() from package MASS, assume a situation with a categorical dependent variable $Y$ with ordered categories $1, \ldots, g, \ldots, k$ and predictors $X_{1}, \ldots, X_{j}, \ldots, X_{p}$. polr() assumes the proportional odds model $$ \text{logit}(p(Y \leqslant g)) = \ln \frac{p(Y \leqslant g)}{p(Y >...


19

In principle you can make the machinery of any logistic mixed model software perform ordinal logistic regression by expanding the ordinal response variable into a series of binary contrasts between successive levels (e.g. see Dobson and Barnett Introduction to Generalized Linear Models section 8.4.6). However, this is a pain, and luckily there are a few ...


16

Here's a little info that might point you in the right direction. Regarding your data, what you have is a response with multiple categories, and anytime you are trying to model a response which is categorical you are right to try and use some type of generalized linear model (GLM). In your case you have additional information which you must take into ...


13

In the case of the multinomial one has no intrinsic ordering; in contrast in the case of ordinal regression there is an association between the levels. For example if you examine the variable $V1$ that has green, yellow and red as independent levels then $V1$ encodes a multinomial variable. If you have a new variable $V2$ were the levels green, yellow and ...


11

Concentrate on a few of the indexes right now. index.orig is the apparent predictive ability/accuracy score when you evaluate it on the data used to fit the model. index.corrected is the cross-validation-corrected version of the same index, i.e., corrected for overfitting (de-biased). Dxy is Somers' $D_{xy}$ rank correlation coefficient - a measure of ...


11

You are making a leap that you need to classify predicted values. The fact that your method never picks the "severe" category is a consequence of the discrete nature of the problem and that "severe" is infrequent. With ordinal response models you can just use exceedance probabilities on their own (for all but one category) or just quote the individual ...


10

Here is an example using the R rms package orm function. The three variables are defined in the original question above. First we see which of 3 families yields the most parallelism. require(rms) row <- 0 for(gvar in list(pred_1, pred_2)) { row <- row + 1; col <- 0 for(fun in list(qlogis, qnorm, function(y) -log(-log(y)))) { col <- col ...


10

You have perfectly confused odds and log odds. Log odds are the coefficients; odds are exponentiated coefficients. Besides, the odds interpretation goes the other way round. (I grew up with econometrics thinking about the limited dependent variables, and the odds interpretation of the ordinal regression is... uhm... amusing to me.) So your first statement ...


9

Lets start with the good news: The proportional odds assumptions does not require that the distances between categories are the same. So what does the proportional odds assumption imply? Say we have three ordered outcomes (1, 2, 3), then we could model the choice of 1 versus 2 or 3 and the choice of 2 versus 3. The proportional odds assumption says that ...


9

This is going to be at best a partial answer but hope it helps a little. Given that your response is ordinal you have to ask yourself whether the distance between different categories is different depending on the starting position. In other words. If you think the gap between 1 and 3 is not necessarily the same gap as the gap between 2 and 4, then using a ...


9

A rank-correlation may be used to pick up monotonic association between variates as you note; as such you wouldn't normally plot a line for that. There are situations where it makes perfect sense to use rank-correlations to actually fit lines to numeric-y vs numeric-x, whether Kendall or Spearman (or some other). See the discussion (and in particular, the ...


9

An ordered logit model is more appropriate as you have a dependent variable which is a ranking, 7 is better than 4 for instance. So there is a clear order. This allows you to obtain a probability for each bin. There are few assumptions that you need to take into account. You can have a look here. One of the assumptions underlying ordinal logistic (and ...


8

Let's think about regular linear regression, and to make it concrete, let's say we are trying to predict height of people. When you regress heights against just an intercept term and no predictors, the intercept term will be be the height averaged over all the people in your sample. Lets call this term $\beta_0^{\text{no predictor}}$ Now, we want to add a ...


8

Ordinal regression is appropriate whenever the dependent variable is ordinal: That is, when you can assume that the levels are in order, but not that the gaps between the levels are equal. If you analyze individual Likert items (each scored 1-5 or 1-7 or whatever) then this could be a very reasonable assumption. But other methods might be good too; this ...


8

Clumping at 0 is called "zero inflation". By far the most common cases are count models, leading to zero-inflated Poisson and zero-inflated negative binomial regression. However, there are ways to model zero inflation with real positive values (e.g. zero-inflated gamma model). See Min and Agresti, 2002, Modelling non negative data with clumping at zero for ...


8

As discussed elsewhere on the site, ordinal regression (e.g., proportional odds, proportional hazards, probit) is a flexible and robust approach. Discontinuities are allowed in the distribution of $Y$, including extreme clumping. Nothing is assumed about the distribution of $Y$ for a single $X$. Zero inflated models make far more assumptions than semi-...


8

You should not dichotomize your dependent variable. You should use ordinal logistic regression, at least as a starting point. You should not remove data.


8

To expand on @Peter Flom's answer: There is almost always more statistical and explanatory power in an analysis that keeps continuous and ordinal variables as such. This effect is larger if you penalize yourself in terms of sample size at the same time. So, let's say you were ignore this advice and go ahead and dichotomize your data, what should you do? ...


7

You would not validly explain the use of OLS: If you interpret the OLS outcome as probability of outcome = 1, then your model says that it is possible for P$(Y=1) > 1$, and for P$(Y=1) < 0$, which is problematic. Your errors are not distributed $\mathcal{N}(0,\sigma)$, thus violating a fundamental OLS assumption. Although you write "None of the ...


7

The problem here is what is called separation. For one level of your predictor (age) you have zero occurrences of one level of your outcome (collagen). The program is driving its estimate of the coefficient upwards to infinity and in this case stopping at an arbitrary value which as you point out when exponentiated gives you a very large odds ratio. What you ...


7

No. The log-likelihood value depends on the scale of the response variable and the size of the dataset. It cannot be meaningfully interpreted in an absolute way.


7

The key thing to consider is the interpretation of each model (more so, in fact, than the proportional odds assumption, which is essentially just a constraint one puts on the parameters in the model). Or, more specifically, "which" logit is being modeled (I'm assuming by "cumulative probability" you refer to the "cumulative logit" models). Let's keep ...


6

The discreteness is not an issue, so much as the ordinal (ordered, graded) scale used for your assessment from normal to severe. That indeed implies something different from standard linear regression, namely some ordinal regression method such as ordered logit or ordered probit. Note incidentally that multivariate regression is not the same as multiple ...


6

There are a variety of ordinal regression models (see Agresti) but they rely on certain assumptions. When those assumptions are violated, the models may become incorrect. The most common assumption is that of proportional odds. Multinomial regression does not make this assumption and can therefore model odds that are not proportional. However, an ...


6

I think it's been asked before. It's useful to realize that, without a prespecified sample size and alpha level, the $p$-value is just a measure of the sample size you ultimately wind up with. Not appealing. An approach I use is this: at what sample size would a 0.05 level be appropriate? Scale accordingly. For instance, I feel the 0.05 level is often suited ...


5

The word "greater" here is poorly defined. The coefficient will tell you the effect of a unit change in the explanatory variable on the linear predictor, $x'\beta$ part of your model. However, if $X_1$ takes values 0 and 1, and $X_2$ varies from -10 to +200, a unit change in the first one is A LOT, and a unit change in the other is minuscule compared to the ...


5

Before answering your question, it is worthwhile to note the difference between individual specific and population averaged models for clustered data, since they are each interpreted differently: Individual specific models explicitly model the dependence at the individual level (e.g. by using random effects). In these models the regression coefficients are ...


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