Statistical Reasoning in Public Health - Math Problem Example

? Statistical Reasoning in Public Health HW#2 Question one a What is the estimated unadjusted mean difference in hourly wages for females as compared to males? Females earn 2.1 dollars less than males per hour when other variable s are not taken into account. 2. Report a 95% confidence interval for this difference. The 95% CI for this difference ranges from -2.98 to -1.22 3. Write a sentence interpreting both the unadjusted mean difference and the corresponding confidence interval. When not considering other variables, women in 1985 earned approximately 2.1 dollars less than the men did; and 95% of the times, the actual values for this estimate would fall between 2.98 dollars to 1.22 dollars less then what the men earned on an average. b. 1. What is the estimated adjusted mean difference in hourly wages for females as compared to males, adjusting for age, union membership, and job type? When age, Union Membership and type of job were taken into consideration, it was found that women earned approximately 1.9 dollars lesser per hour then men did in 1985. 2. Report a 95% confidence interval for this difference. The 95% CI for this difference ranges from -2.76 to -1.04. 3. Write a sentence interpreting both the adjusted mean difference and the corresponding confidence interval. When considering other variables like age, union membership and job type, women in 1985 earned approximately 1.9 dollars less than the men did; and 95% of the times, the actual values for this estimate would fall between 2.76 dollars to 1.04 dollars less then what the men earned on an average. 2. Comment on any disparities in the estimated mean difference in hourly wages between males and females in the four models whose results are listed above. Does it appear from these results that the wage/gender relationship in confounded by other worker characteristics such as worker age, membership in a union, and job type? Why or why not? Across the four models; it is evident that woman did earn lesser than men did per hour in 1985. This effect was seen regardless of the variables included in the study. Adjusting for age increased the discrepancy slightly; while adjusting for job type and union membership in addition to age decreased the discrepancy between the average wages of men and woman slightly. In spite of this, it is difficult to say that these variables play a very significant role; since the differences seen were quite small; and the confidence intervals for the four models overlapped quite a lot. It may be said that variable like age, union membership and job type did affect the discrepancy in the average wages of men and women; but this effect was quite small. 3. Use the results from Model D to estimate the mean difference in hourly wages for females, age 42, who are union members with manufacturing jobs, as compared to 42-year male union members with manufacturing jobs. (Note that you have already done this in a previous portion of the problem – I am just trying to “drill into you” how to interpret multiple linear regression coefficients.) After controlling for Union membership, type of job and age, it was found that women earned on an average, 1.9 dollars less than men in 1985. 4. 1. Does the given information allow you to assess whether the relationship between hourly wages and sex is modified by age? By itself, the data available is not enough to assess if the relationship between hourly wages and sex is modified by age. 2. If not, what additional results would you need to see? We would need information on the significance testing of the F values for the regression analyses in order to estimate whether age does truly affect the relationship between hourly wages and sex; or whether the observed effect is simply a function of the sampled data. 2. Question two a) 1. What is the estimated adjusted association between Diastolic Blood Pressure (DBP) and the snoring index (SI)? (recall the snoring index ranges from 0-1800+ in the study) The estimated adjusted association between Diastolic Blood Pressure (DBP) and the Snoring index (SI) is 0.000 – which shows a zero rise in DBP for each unit change in SI. This effect was not significant. 2. Give a 95% CI for the population level association between DBP and the SI. The 95% CI for the population level association between DBP and the SI ranged from -0.002 to 0.002 units of changes in DBP per unit change in SI. This effect was not significant. b) Do you think it is reasonable to assume a linear relationship between DBP and BP across the entire range of SI values? What graphical information would be helpful in investigating this? From the presented data, it is not possible to deduct the relationship shared by DBP and BP as one moves across the values of SI; although the trends seem to be similar, and an assumption may be based on this data. If the scores for DBP and BP were plotted in a scatter plot for all participants with SI on the x axis, and the lines of regression were obtained for each; then it would be possible to estimate the relationship between these variables. c) 1. For both systolic blood pressure and diastolic blood pressure outcomes do men or women tend to have higher blood pressures on average? Men tend to show higher bold pressure than women as measured for both systolic blood pressure and diastolic blood pressure. This is seen for adjusted as well as unadjusted data. 2. How did you use the regression results to answer this? The regression results reported state that they are for men, meaning that men should be coded as 1 and women as 0. The results show that there is increase in blood pressure as we move from code 0 to code 1; showing that men tend to have higher blood pressure. For both systolic blood pressure and diastolic blood pressure, these effects are statistically significant; showing that the effect may be generalized to the population. d) Suppose your classmate argues that any differences in the systolic blood pressure (SBP) average between men and women is really because of other differences between men and women that are related to systolic blood pressure: specifically BMI and snoring index. Do the results support your classmate’s statement? Why or why not? It is difficult to support such an argument – “that any differences in the systolic blood pressure (SBP) average between men and women is really because of other differences between men and women that are related to systolic blood pressure” on the basis of SI, as SI has been found to be ineffective in predicting SBP when adjusted for other variables. Although common sense tells us the BMI would be different for males and females; and could thus play a role in the difference between the exhibited levels of SBP; we do not have data that shows us the nature of the relationship shared by sex and BMI particularly. Without this data, such an argument cannot be supported. e) Can these results, as given, be used to assess whether the relationship between SBP and snoring (as measured by the SI) is different for men and women? (In other words, is the association between SBP and SI modified by sex)? Why or why not? With only the present data, it is not possible to verify if the relationship between SBP and snoring (as measured by the SI) is different for men and women; since data about the relationship between SI and sex is not presented. It would be possible to estimate if sex affects the relationship between SBP and SI only if data that controlled sex while predicting the effect of SI on SBP were provided free of other possible confounders. 3. Question Three a) According to the table footnotes, what unit did the authors use for age in the multiple linear regressions? The authors used years in five-unit changes to measure age. The regression coefficient gives the changes in IMT per 5 years for the subjects. b) What is the estimated mean difference in IMT for two groups of persons who differ by one-year in age, adjusted for the other predictors in the model? The study data has been analyzed for groups that are on an average 5 years apart. To estimate the difference between groups that are one year apart; the data needs to be analyzed accordingly. A tentative assumption that the difference in IMT would be one fifth of what was observed in the study data may be proposed; which would bring the mean difference in IMT between groups of individual a year apart to 0.026 / 5 = 0.0052. This value should not be accepted until it is obtained through systematic analysis. c) Estimated a 95% CI for the quantity estimated in part b. The CI calculated and reported for this study is for a 5 unit change in age. It is difficult to estimate the CI for one year on the basis of this value. In order to obtain a CI for the said value; data for groups one year apart should be analyzed; and the SE calculated for the same. The obtained value should be used to obtain the CI. d) Interpret the slope of sex in words. The regression coefficient for IMT on sex is 0.009; and has been found to be significant at the 0.02 level (p < 0.02). This means, that men on an average, show 0.009 mm more IMT then do women. e) Give a 95% CI for the slope of sex The 95% CI for the regression of IMT on sex ranges from 0.001mm to 0.017mm. f) What additional information would you need to assess whether the relationship between IMT and sex is confounded by at least some of the additional predictors from the given multiple linear regression? We would require the un-adjusted values of the regression of IMT on sex. If there is a particular difference in the two values – adjusted and unadjusted; it would mean that other factors do affect the relationship between IMT and sex. g) What additional information would you need to assess whether sex modifies the relationship between IMT and smoking, after adjusting for age, LDL, BMI and SBP? We would require the adjusted values of the regression of IMT on smoking that control for LDL, BMI and SBP; but not sex. If there is a particular difference in the two values; it would mean that sex does affect the relationship between IMT and smoking keeping other factors constant. h) Given the above results, can you ascertain whether the linear relationship between IMT and the six predictors in the regression is strong? Why or why not? With the given data, it is possible to determine that IMT has a strong linear relationship with the six variables discussed by the study. This is evident since the adjusted coefficients obtained for linear relationships are all significant; though the coefficient with LDL is significant at 0.06 level of significance; just short of the accepted 0.05 level. Thus, the linear relationships IMT has with each of these variables are applicable to the population. The strength of the relationships, on the other hand, cannot be ascertained without the r and r2 values for these relationships. i) Suppose above results are used to compare average IMT between 39 year olds to 29 year old after adjusting for the other 5 predictors in the model– what would be the estimated average difference in IMT? Compute a 95% confidence interval for this difference. In this study, the regression of IMT on age has been calculated for groups that are roughly five years apart. For groups that are ten years apart; the mean difference may be proposed to be roughly double of the observed value –i.e. 0.052mm. This is a rough estimate that may be used until data is collected and analyzed; as the individuals represent the ages that are the end points of the population tested in this study. A true estimate would be required to verify the validity of this supposition. Also; it would be inappropriate to obtain a CI for this value using the SE calculated on the present data. j) Would it be appropriate to use the above results to estimate the adjusted average difference in IMT levels for 80 year olds compared 70 year olds? Why or why not? It would not be appropriate to use the study results to estimate the adjusted average difference in IMT levels for 80 year olds compared 70 year olds. For one, the above value is a rough estimate; and not a verified estimate. Secondly; there are likely to be different factors affecting IMT for 70 and 80 year old individuals as compared to 29 and 39 year old individuals. Since age has been seen to have a significant relationship with IMT in this study, this concern is further strengthened. 4. Question four a. Graph (or review the posted graph) the relationship between TLC and age in a Scatter plot. Comment on the nature of the relationship between TLC and age. Based on this scatter plot; there seems to be a weak relationship – if any shared by TLC and age. Any relationship that does exist seems to be a weak, positive relationship. b. Perform (or review the posted Stata output) a simple linear regression of TLC on age. 1. Are the results consistent with what you saw in the scatter plots? Yes, the results seem consistent with the observed scatter plots, since the obtained coefficient was a low positive one; and the regression value was not statistically significant, showing that it was not possible to predict TLC from age. 2. Report and interpret the estimated coefficient (slope) of age in a sentence. The obtained coefficient of age was 0.0358637, not significant. This shows that it is not possible to predict TLCV from age or participant. 3. Report a 95% confidence interval for the (true) coefficient of age for this population. The 95% CI was found to range from -.020114 till .0918415. c. Graph the relationship between TLC and height in a scatter plot. Comment on the nature of the relationship between TLC and height. The variable of TLC and height seem to share a direct (positive) relationship such that scores on TLC rises with increase in height. c. Perform a simple linear regression of TLC on height. 1. Are the results consistent with what you saw in the scatter plots? Yes, the regression analyses conforms the data obtained from the scatter plot. 2. Report and interpret the estimated coefficient (slope) of height in a sentence. The coefficient of regression is .0934323; and the t value obtained is 5.46, which was significant at the 0.0001 level of significance (p < 0.001). This shows that height and TLC share a strong relationship, and it is possible to predict TLC from height such that for every 1 unit increase in height, there was a corresponding 0.0934323 unit’s increase in TLC. 3. Report a 95% confidence interval for the (true) coefficient of height for this population. The range for the 95% CI for height was found to be from 0.0585055 to 0.1283591 e. Graph the relationship between TLC and patient’s sex in a scatter plot. 1. Is this a useful exploratory approach for assessing gender differences in TLC? A scatter plot is not a very useful technique when one of the variables is a categorical, binary variable. 2. Can you suggest another way of exploring the relationship between a continuous outcome and a binary predictor? Rather than a scatter plot; a box-plot would be more useful in completing this analysis. f. Perform a simple linear regression of TLC on sex. 1. Report and interpret the estimated coefficient (slope) of sex in a sentence. The coefficient for regression of TLC on sex was -1.77875, and the t value was -3.67 which was found to be significant at the 0.001 level (p < 0.001). This means that it is possible to predict TLC from sex, and females are associated with lower TLC then men. 2. Report a 95% confidence interval for the (true) coefficient of sex for this population. The range for the 95% CI for sex was found to be from -2.768782 to -.7887183. g. Now perform a multiple linear regression of TLC on height, age, and sex together. Report and interpret the slope estimates for: 1. Height The multiple regression coefficient for TLC on height was .0910637, and the t value was 3.81 which were found significant at the 0.001 level. (p < 0.001). this shows this shows that height and TLC share a strong relationship, and it is possible to predict TLC from height such that for every 1 unit increase in height, there was a corresponding .0910637 unit’s increase in TLC when age and sex are kept constant. 2. age The multiple regression coefficient for TLC on age was -.0275141, and the t value was -1.18; which was found to be not significant. This shows that TLC cannot be predicted from age, if height and sex are kept constant. 3. sex The multiple regression coefficient for TLC on sex was -.6361484, and the t value was -1.28; which was found to be not significant. This shows that TLC cannot be predicted from sex, if height and age are kept constant. h. Which predictors are statistically significantly associated with TLC (?=.05) in the multiple linear regression model? Height is the only predictor that is significantly associated with TLC in the multiple linear regression model. i. Compare the unadjusted relationship between TLC and sex, to the height and age adjusted association between TLC and sex. Is there any suggestion of confounding? Why/why not? The unadjusted relationship between TLC and sex is significant, while the adjusted one is not. This shows that there is a likelihood of confounding of height (the only significant predictor in the multiple regression model) in the unadjusted relationship. This can be confirmed by testing the relationship between height and sex. j. 1. What is the R2 value for the multiple regression model you fit in (g)? The unadjusted R2 value for the multiple regression model is 0.5554; while the adjusted R2 value for the multiple regression model is 0.5078. 2. what is the interpretation of this value? The value of the R2 is an estimate of the percentage of TLC values explained by the age, height and sex of the individual together. k. Using the regression model results from part (g), estimate the mean TLC level for: 1. 42 year old males, 170 cm tall Since age and sex seem to have no significant relationship with TLC, the calculation should be done on the basis of height only. The TLC is roughly about 8 ltr. 2. 35 year old females, 145 cm tall Since age and sex seem to have no significant relationship with TLC, the calculation should be done on the basis of height only. The TLC is roughly about 3.75 ltr. l. Using the regression model results from part (g), estimate the mean difference in TLC between 50 year old females 150 cm tall, and 40 year old males 160 cm tall, Since neither sex nor age are significant predictors of the TLC levels exhibited by an individual; the difference in TLC of individuals may be predicted on the basis of height alone. For each cm increase in height; the TLC increase seen is 0.0910637. thus, the difference in TLC between an individual 150 and 160 cm tall would be roughly 0.92 liters. m. Create (or review the posted) a scatter plot of TLC versus height separately for male and for females. Does the relationship between TLC and height appear similar for both sexes? The relationship shared by TLC and height seems to be different for the two sexes. For men, as height increases; TLC seems to reduce as a function of height after peaking at close to a height of 170 cm. on the other hand, for females, the relationship between TLC and height seems to be a direct relationship with increase in TLC as height increases. On the other hand, the data on men has very few cases below 170 cm height – which show lower TLC, while the data on females have only a couple of cases above 170 cm – both showing lower TLC as compared to those with height close to 170 cm. thus, the trends may actually be similar for both males and females – with TLC rising with height till 170cm, and then dropping as a function of increase in height beyond 170cm. n. Run a regression of TLC on height, sex, and an interaction between sex and height. Based on this result: 1. What is the estimated mean difference in TLC for two groups of men who differ by 1 cm in height? Calculating the mean difference in TLC for two groups of men who differ by 1 cm in height would not be appropriate with this data, since the interaction is not significant. 2. What is the estimated difference between two groups of women who differ by 1 cm in height? Calculating the mean difference in TLC for two groups of women who differ by 1 cm in height would not be appropriate with this data, since the interaction is not significant. 3. Is there a statistically significant interaction between sex and height? The interaction between sex and height does not seem to be statistically significant as per the data provided. 5. Based on these results, what is the estimated mean difference in medical home scores for practices with a pay for performance index of 3, compared to practices with a pay for performance index of 1, adjusted for all other factors used in the multiple regression? a. (Adjusted) Mean differences cannot be estimated from multiple linear regressions. b. 3.24% c. 6.48% d. 9.72% 6. What is the estimated adjusted mean difference in medical home scores for practices with 13- 19 physicians compared to practices with 8-12 physicians? a. (Adjusted) Mean differences cannot be estimated from multiple linear regressions. b. 8.86 % c. 8.86% - 5.73 %= 3.13% d. 5.73% -8.86% = -3.13% 7. What additional information would you need to see to determine if the relationship between medical home scores and practice size is confounded by at least some of the additional factors included in this multiple linear regression analysis? a. No additional information is necessary to address this. b. The unadjusted relationship between medical home scores and practice size. c. Separate regression results relating medical home scores and practice size for practices owned by a hospital, health system or HMO, and practices owned by physicians d. No additional information will be helpful, as regression cannot be used to assess confounding. Read More