## Tuesday, October 23, 2012

### Ordinal Regression Probability Ratio

One of the statistical techniques I've proposed to use in my dissertation proposal is ordinal regression (also known as ordered logit, proportional odds model, and cumulative odds model).  This regression model is a direct extension of the binary logistic model except that instead of modeling the probability of a binary event (e.g. alive/dead), you are modeling the probability of an inequality with three or more naturally ordered events (e.g. mild/modest/severe).  Consider the binary logit model:
$\frac{Pr(y = 1 \vert {\boldsymbol{X=x}})}{Pr(y = 0 \vert {\boldsymbol{X=x}})}$
Now consider the ordinal logit model:
$\frac{P(Y \geq y_i \vert {\boldsymbol{X=x}})}{1 - P(Y \geq y_i \vert {\boldsymbol{X=x}})} = \frac{P(Y \geq y_i \vert {\boldsymbol{X=x}})}{P(Y < y_i \vert {\boldsymbol{X=x}})}$
(If interested in some background concerning the logit model, you can find a mapping of the inverse logit to the logistic model here, the logit model likelihood function here, and the logit model maximum likelihood estimates here.)

By definition (and derivation) we have this,
$Pr(Y \geq y_i \vert {\boldsymbol{X=x}}) = \frac{1}{1 + e^{-(\alpha_i + {\boldsymbol{x'_i \beta}})}}$
and this,
$Pr(Y < y_i \vert {\boldsymbol{X=x}}) = \frac{1}{1 + e^{\alpha_i + {\boldsymbol{x'_i \beta}}}}$

Since the ordinal regression model is the ratio of the two probabilities immediately preceding, the following is obtained after substitution and some minor manipulation:
$\frac{P(Y \geq y_i \vert {\boldsymbol{X=x}})}{P(Y < y_i \vert {\boldsymbol{X=x}})} = \frac{1 + e^{\alpha_i + {\boldsymbol{x\beta'}}}}{1 + e^{-(\alpha_i + {\boldsymbol{x\beta'}})}}$
which then reduces to $e^{\alpha_i + {\boldsymbol{x\beta'}}}$.

When I was reviewing this a couple of weeks ago, though, I wasn't able to remember how the ratio of the two exponential expressions reduced to the single exponential expression.  Embarrassingly, I mentioned it to my wife (a wicked smart chick with an applied math background) and she thought about it for all of ten minutes then scribbled the solution onto a piece of newspaper.  Although not obvious to me then, the solution seems so obvious to me now:

$\frac{1 + e^{\alpha_i + {\boldsymbol{x\beta'}}}}{1 + e^{-(\alpha_i + {\boldsymbol{x\beta'}})}} \frac{e^{\alpha_i + {\boldsymbol{x\beta'}}}}{e^{\alpha_i + {\boldsymbol{x\beta'}}}} = \frac{e^{\alpha_i + {\boldsymbol{x\beta'}}}(1 + e^{\alpha_i + {\boldsymbol{x\beta'}}})}{e^{\alpha_i + {\boldsymbol{x\beta'}}}(1 + e^{-(\alpha_i + {\boldsymbol{x\beta'}})})} = \frac{e^{\alpha_i + {\boldsymbol{x\beta'}}}(1 + e^{\alpha_i + {\boldsymbol{x\beta'}}})}{(1 + e^{\alpha_i + {\boldsymbol{x\beta'}}})} = e^{\alpha_i + {\boldsymbol{x\beta'}}}$

## Wednesday, October 17, 2012

### Optimism: will good things follow?

I defend my dissertation proposal in a few weeks and as I read and review the literature on optimism and pessimism I'm struck by not only the breadth, but the depth of the literature.  I remarked to my wife last night that it feels as if I'm working my way down a wormhole:  it twists, it turns, and just as it looks like I've reached the bottom it continues on.  A person could conceivably spend several weeks doing nothing but searching, reading, and reviewing articles and books dealing with optimism-pessimism, although this may be a questionable use of time and resources since, according to Ebel, Bliefert, & Russey in "The Art of Scientific Writing", the number of sources retrieved for a background/lit review in a dissertation/thesis should be capped at around fifty.  I'm well past that but no sense on dwelling on it now.  At any rate, as part of my literature review and background of my proposal, I've read many papers from the scientific literature as well as a few mainstream, pop psychology books.  There isn't, of course, a consensus on whether being dispositionally optimistic is uniformly beneficial in all settings and circumstances.  If there were then there wouldn't be much point in me pursuing my dissertation question:  does increased dispositional optimism improve medication adherence in a population of people living with HIV/AIDS?  Many of the findings from the scientific literature suggest that being predisposed to optimism affords a person several advantages:  less depression, more active coping strategies, less anxiety, and better health outcomes.  But there are enough papers showing negative or null effects of optimism on health to cast just enough doubt such that the central question --- is optimism good for you? --- remains unresolved.  In the matrix below is a sampling of papers from the scientific literature (an exhaustive list of papers would necessitate a book, not a blog post).  As is evident, more papers than not report positive effects from optimism but this tendency is far from robust.

As for pop psychology books, the most recent one I read, "Breaking Murphy's Law:  How Optimists Get What They Want from Life --- and Pessimists Can Too" by Dr. Suzanne Segerstrom was decent but suffered from an identity crisis.  The schizophrenia aside, though, Segerstrom marshals the evidence from the scientific literature as well as relies on her research to support her thesis that optimism is a good thing because optimistic people behave in particular ways, e.g. they are more engaged, more focused on goals, and utilize active coping strategies more frequently than their pessimistic brethren.

On the other end of the optimism-is-good-for-you mainstream books lies Barbara Ehrenreich's "Bright-Sided:  How Positive Thinking is Undermining America".  Unlike Segerstrom's book, Ehrenreich argues that the widespread unerrant positive thinking characteristic of America is tantamount to mass delusion.  She describes her introduction to optimistic thinking by way of a breast cancer diagnosis and how the current of "think positively" was distracting, discouraging, and detracted from the reality of her diagnosis.  She doesn't limit her investigation of optimistic thinking to cancer, however, Ehrenreich also discusses how optimism relates to wealth, business, religion, and the implosion of the American economy.  In the end, Ehrenreich advocates for less bright optimism and for more realism, skepticism, and critical thinking.

In Tali Sharot's "The Optimism Bias:  A Tour of the Irrationally Positive Brain", there is less polemic and more explanation.  Optimism bias, she explains, is defined as "the inclination to overestimate the likelihood of encountering positive events in the future and to underestimate the likelihood of experiencing negative events" (pp. xv).  Put more simply, it is the tendency to have expectations that are slightly better than what the future holds.  Sharot discusses many situations and settings where optimism bias is present and eventually concludes (with the help of academics cited in her book) that "optimism is like red wine:  A glass a day is good for you, but a bottle a day can be hazardous" (pp. 198).

So how does all this square with my dissertation?  Well, I'm not sure yet but following approval of my proposal, I'll find out.