Update writeup.md

rat/eamentalhealth/analysis/writeup.md

@@ -372,7 +372,7 @@ Of course, the relation "(significantly) correlated to" is not transitive in the
 What is the effect of mental health on productivity, and how does access to healthcare mediate it? 
 
 #### 9.1. How does mental health affect productivity?
-We can correlate our four markers of mental illness on our two measures of lost productivity lost. However, the correlation runs both ways: from the data we cannot deduce whether people with mental conditions lose productivity, or whether losing productivity (i.e., losing a job) makes people more likely to be mentally ill. 
+We can correlate our four markers of mental illness on our two measures of lost productivity. However, the correlation runs both ways: from the data we cannot deduce whether people with mental conditions lose productivity, or whether losing productivity (i.e., losing a job) makes people more likely to be mentally ill. 
 
 If we assume that the effect is purely unidirectional (mental illness -> lost productivity), three highlights are:
 - Being diagnosed with one or more mental conditions would cause a loss of ~9 hours per 2 weeks.
@@ -566,6 +566,7 @@ In this last case, the implied effect doubles, to 2.2 hours saved per week, as s
 
 ```
 > m_ill3= (Receiving_positive | Receiving_negative) & A$m_ill_or_not
+## Here, we restrict the regression to people diagnosed with a mental condition.
 > sum(m_ill3)
 [1] 124
 > summary(lm(hours_lost[m_ill3] ~ Receiving_positive[m_ill3]))
@@ -581,9 +582,12 @@ Receiving_positive[m_ill3]TRUE   -4.428      4.018  -1.102    0.273
 Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 ```
 
-Note that, in none of the above cases we've gotten a significant effect. This is because outliers, for example, respondents who lost a job because of mental illness, and thus ~40 hours per week, make the error bars huge. In this case, excluding outliers doesn't make sense at all; instead, knowing where the error comes from, I think it makes sense to not get hung up on p-values. As a proof of concept, here is the same regression as above, but considering the logarithm of hours lost (which gives less weight to outliers); we "suddenly" reach significance (p-value of 0.00698). On the flip-side, the logarithm of hours lost is not an intuitive unit in which to report results. 
+Note that, in none of the above cases we've gotten a significant effect. For example, if we consider the whole population, our coefficients for receiving and not receiving satisfactory mental healthcare are significant, but not significantly different from each other.
+
+I think that this is a function of outliers. For example, respondents who lost a job because of mental illness, and thus ~40 hours per week, make the error bars huge. In this case, excluding outliers doesn't make sense at all; instead, knowing where the error comes from, I think it makes sense to not get hung up on p-values. As a proof of concept, here is the same regression as above, but considering the logarithm of hours lost (which gives less weight to outliers); we "suddenly" reach significance (p-value of 0.00698). On the flip-side, the logarithm of hours lost is not an intuitive unit in which to report results. 
 
 ```
+## Here, we restrict the regression to people diagnosed with a mental condition.
 > summary(lm(log(hours_lost[m_ill3]+1) ~ Receiving_positive[m_ill3]))
 Coefficients:
                                Estimate Std. Error t value Pr(>|t|)    
@@ -655,7 +659,7 @@ Overall_work_impairment = Percent_missed_due_to_mental_health + (1-Percent_misse
 Overall_work_impairment = Overall_work_impairment*100 ## To express this is percentages.
 ```
 
-We will use the percent overall work impairment due to health as our productivity measure.
+We will use the percent overall work impairment due to mental health as our productivity measure.
 
 Now, we can correlate productivity lost with having or not having a mental illness. Because I'm not sure respondents understood that an answer of 5 on a scale of 1-10 would be interpreted as a 50% reduction in effectiveness, I'm hesitant to interpret this as a percentage. If we speak about points in the abstract:
 - When regressing lost productivity on mental conditions diagnosed and intuited: Being diagnosed with a mental condition is correlated with 42 points of lost productivity, and Intuiting one has a mental illness (as opposed to having been diagnosed with one) is correlated with a loss of ~27 points of lost productivity
@@ -796,6 +800,7 @@ F-statistic: 91.34 on 2 and 268 DF,  p-value: < 2.2e-16
 
 #### 9.4. Correlation of the Work Productivity and Impairment Scale with mental health, mediated by an index of access.
 With respect to accessibility, we can consider the following 4 questions:
+
 14. How challenging was it to receive the mental healthcare services you needed within the past 12 months?    
 15. How challenging is it to find useful information on mental healthcare services?     
 16. Do you experience financial difficulties as a result of mental healthcare?    
@@ -904,7 +909,7 @@ However, if taken at face value, these answers imply that the value of providing
 
 Using the same methodologies as above, the crossectional estimate of providing better information is also large. A productivity improvement of ~10% if information is very easy to come by, respectively a productivity loss of ~10% if finding it is very hard (~12% if restricting the regression to those with a diagnosis)
 
-Alone in terms of work hours, 2 would be gain (resp. lost) every two weeks if respondents find information very easy to come by (resp. very hard), amongst respondents who have been diagnosed with a mental illness or think they have one (~3 hours if one only considers those with a diagnosis).
+Alone in terms of work hours, 2 would be gained (resp. lost) every two weeks if respondents find information very easy (resp. very hard) to come by, amongst respondents who have been diagnosed with a mental illness or think they have one. If we only look at respondents with a diagnosis, this estimate increases to ~3 hours.
 
 I personally consider it likely that the effect is so large because the causal mechanism goes in both directions: a less burdensome mental illness -> easier to do things like finding information, or not missing work hours, but also: information is easier to find -> condition gets better -> fewer work hours are missed.