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. 

Author(s); (year)
Positive, Negative?
Summary/Main Finding(s)
Non-HIV/AIDS Conditions
Allison, Guichard, et al.; (2003)
Dispositional optimism predicts survival status 1 year after diagnosis in head and neck cancer patients
The LOT was translated into French and the authors followed head & neck cancer patients for the first year following diagnosis.  The multivariate analyses indicated that those who were pessimistic (continuous metric) were 12% more likely to be dead at one-year (OR=1.12, 95% CI 1.01-1.24).
Allison, Guichard, et al.; (2000)
A prospective investigation of dispositional optimism as a predictor of health-related quality of life in head & neck cancer patients
Optimists and pessimists categorized according to score on LOT --- those above median deemed ‘optimists’ and those below deemed ‘pessimists’.  HRQOL measurements made at baseline and three months later w/ optimists rating HRQOL better than pessimists at both time points.  Optimists appear to fare better w/ respect to role & cognitive functioning, less pain and better HRQOL both before and after treatment.
Aspinwall, Taylor; (1992)
Modeling Cognitive Adaptation:  A longitudinal investigation of the impact of individual differences and coping on college adjustment and performance
Positive (qualified)
Optimism exerted a direct, positive effect on subsequent adjustment to college but much of this effect was mediated by coping methods.
Carver, Lehman, Antoni; (2003)
Dispositional pessimism predicts illness-related disruption of social and recreational activities among breast cancer patients
Positive (qualified)
Pessimism predicted higher levels of illness-related disruption in social and recreational activities as well as higher levels of emotional distress and fatigue during year after surgery.  The effects were mediated by distress and fatigue, however.
Carver, Pozo-Kaderman, et al.; (1994)
Optimism versus pessimism predicts the quality of women’s adjustment to early stage breast cancer
Pessimists adjusted more poorly to adverse psychosocial changes.  The sense of pessimism about one’s life enhances risk for adverse psychological reactions to the diagnosis and treatment of breast cancer.
Carver, Smith, et al.; (2005)
Optimistic personality and psychosocial well-being during treatment predict psychosocial well-being among long-term survivors of breast cancer
Investigators examined how optimism-pessimism (from the LOT) influenced well-being over a 5-13 year time frame.  Optimism maintained statistical significance (tempered somewhat in the multivariate setting) with respect to long-term psychological well-being.  Side note:  O-P was repeatedly measured and shown to be stable.
Carver, Gaines; (1987)
Optimism, pessimism, and postpartum depression
Study found that optimism associated (correlated) with resistance to development of postpartum depression at three weeks post-childbirth.  O-P measured via LOT.
Friedman, Nelson, et al.; (1992)
The relationship of dispositional optimism, daily life stress, and domestic environment to coping methods used by cancer patients
Dispositional optimism (via the LOT) was related negatively to avoidance coping but related positively to active-behavioral coping.  Findings suggest that optimists behave (cope) in ways that invite more positive outcomes.
Marshall, Lang; (1990)
Optimism, self-mastery, and symptoms of depression in women professionals
Optimism, as measured by the LOT, overlaps substantially with self-mastery although they are empirically distinct.  Both measures associated with depressive symptoms although only self-mastery associated with symptom levels. 
Mazanec, Daly, et al.; (2010)
The relationship between optimism and quality of life in newly diagnosed cancer patients
Authors examined dispositional optimism (via LOT-R) in population of newly diagnosed cancer patients and found that optimism correlated with anxiety, depression, and spirituality although it did not reach statistical significance in any of the multivariate models (overall HRQOL, physical/social/emotional/functional well-being).
Moyer, Fontana, et al.; (2003)
The role of optimism-pessimism in HRQOL in chronic hepatitis C patients
Positive (qualified)
The LOT was *not* used for the measure of optimism-pessimism so any interpretation should be qualified.  With that said, the findings showed that pessimists exhibited lower QoL of scores and pessimism adversely associated with coping style, overall emotional well-being, and health status in a population of chronic hepatitis C patients.
Schou, Ekeberg, et al. (2004)
Pessimism as a predictor of emotional morbidity one year following breast cancer surgery
Dispositional optimism (via LOT-R) was bifurcated at 18 (<18 pessimist; >=18 optimist) with findings indicating a strong association between LOT score and anxiety and depression at each time point.  Optimism also statistically significant in two multivariate logistic regression models examining anxiety & depression, respectively (optimism confer a beneficial effect).
Schofield, Ball, et al.; (2004)
Optimism and survival in lung carcinoma patients
In a prospective study comparing two treatments for lung carcinoma the investigators administered the LOT at both baseline and post-treatment (six weeks post) and observed a slight decrease (stat significant) decline in optimism between time points as well as a mostly overlapping survival experience (progression-free and overall) among the three arbitrarily selected optimism groups (low, medium, high).  Study adjusted and controlled for some confounders, e.g. performance status, age) but not all, especially depressive symptoms and affectivity (self-efficacy).  Authors conclude that optimism afford no discernible benefit to cancer patients.
Schulz, Bookwala, et al.; (1996)
Pessimism, age, and cancer mortality
Negative (in that optimism didn’t positively affect mortality)
Paper investigating the effects of pessimism, optimism, depression, cancer site, and symptomology on mortality in a sample of 200+ cancer patients.  Optimism and pessimism obtained from the LOT.  Pessimism and depression were significantly positively correlated and pessimism and optimism were significantly negatively correlated.  They conducted hierarchical logistic regression analysis and observed non-significance for main effects of optimism and pessimism on mortality but a statistically significant interaction between age and pessimism.  Younger subjects (30-59) who died exhibited higher pessimism than all other subjects.  Take home message is that pessimistic life endorsement is a risk factor for mortality but only among the younger subjects (30-59).     
Thomas, Britt, et al.; (2011)
Dispositional optimism buffers combat veterans from the negative effects of warzone stress on mental health symptoms and work impairment
A cross-sectional study examining how dispositional optimism (as measured by the LOT-R) moderates the relationship between both acute and chronic warzone stressors and PTSD and depressive symptoms and, also, whether optimism moderates the relationship between mental health symptoms and work impairment.  The authors conclude:  “We found that dispositional optimism buffered both warzone stressors on PTSD symptoms but only deployment demands on depressive symptoms.  The moderating role of dispositional optimism followed the same pattern:  soldiers higher in dispositional optimism reported fewer mental health symptoms when reporting higher levels of warzone stressors compared with soldiers lower in dispositional optimism.  Similarly, soldiers higher in dispositional optimism were less likely than those lower in optimism to report work impairment when experiencing higher levels of PTSD and depression symptoms”
Holmes, Pace; (2002)
HIV-seropositive individuals’ optimistic beliefs about prognosis and relation to medication and safe sex adherence
Investigators sought to determine if women, minorities, and IDUs are as optimistic about their prognosis as other AIDS populations.  Optimism was crudely defined based on expectation of how long the subject expects to live with the results suggesting that those who don’t take their medications were more pessimistic about their life expectancy.
Moyer, Ekpo, et al.; (2008)
Quality of life, optimism/pessimism, and knowledge and attitudes toward HIV screening among pregnant women in Ghana
Negative / Indeterminate
Optimism (per the LOT-R) not associated w/ basic demographics, self-related (mental) health status, QoL, or HIV acceptance.  Curiously, those with the highest levels of optimism had the least amount of HIV knowledge or had not been HIV-tested. 
Question to consider:  Are optimists in denial about risks of disease/hardship they encounter?
Somlai, Kelly, et al. (2000)
Life optimism, substance use, and AIDS-specific attitudes associated with HIV risk behavior among disadvantaged innercity women
The measure of optimism in this study appears to have been created by the authors so any inference is suspect.  That aside, bivariate analyses suggest that higher risk women had lower levels of personal life optimism and in the multivariate analysis (forward stepwise regression) one of the four significant variables was optimism thus suggesting that those w/ lower optimism were more likely to be high-risk for HIV infection.
Taylor, Kemeny, et al.; (1992)
Optimism, coping, psychological distress, and high-risk sexual behavior among men at risk for AIDS
Two types of optimism examined – dispositional and event-based (AIDS) – with the former being derived from the LOT with respect to threat of AIDS infection.  Authors found that dispositional optimism associated with less distress, less avoidant coping, positive attitude as coping strategy, and fewer-AIDS related concerns. 
Milam, Richardson, et al.; (2004)
The Roles of Dispositional Optimism and Pessimism in HIV Disease Progression
Authors examined optimism and pessimism as separate unipolar constructs (as derived from the LOT-R) in relation to disease progression (CD4 count and viral load, respectively).  Bivariate analyses conducted between optimism, pessimism, the dependent variables, and an array of independent variables.  Regression analyses found that higher pessimism associated w/ higher viral load and that moderate optimism associated w/ higher CD4 count.  Per the article, “In summary, the results suggest that low levels of pessimism or moderately high levels of optimism may protect HIV+ persons on ART from progression of disease in the short term.” 
Tomakowsky, Lumley, et al.; (2001)
Optimistic Explanatory Style and Dispositional Optimism in HIV-infected men
Indeterminate / Mildly Positive
Authors sought to investigate effects of optimistic explanatory style and dispositional optimism (from LOT) on HIV symptoms and immune status (CD4 count), respectively.  In multivariate regression models, optimistic explanatory style significantly associated w/ fewer HIV symptoms whereas dispositional optimism moderately associated.  Dispositional optimism not associated w/ CD4 count although optimistic explanatory style related to worse immune function both in cross-sectional and prospective analysis. 
Reed, Kemeny, et al.; (1994)
Realistic acceptance as a predictor of decreased survival time in gay men with AIDS
Negative / Indeterminate
Overarching objective of study was to assess relationship between “realistic acceptance” and survival time so dispositional optimism (via LOT) was tangential to analysis.  That caveat aside, realistic acceptance was not associated with dispositional optimism.  Authors conclude that subjects w/ low realistic acceptance scores survive longer than subjects with high realistic acceptance scores. 
Ironson, Balbin, et al.; (2005)
Dispositional optimism and the mechanisms by which it predicts slower disease progression in HIV:  proactive behavior, avoidant coping, and depression
Prospective study indicates that dispositional optimism (composite of LOT & LOT-R) predicts slower progression of two markers of disease progression:  CD4 count and viral load.  This association, however, was tempered by the inclusion of depression, avoidant coping, and proactive behavioral variables since optimists were more proactive, less depressed, and relied less on avoidant coping.