OSH/5

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Content warning: in this OSH, we present data from the BC coroner's office about overdose fatalities.
Q3 has nothing to hand in.

The opioid crisis is taking a heavy toll on British Columbia. Trends change with many variables, including economic conditions, public policy, and time. In this OSH, we investigate how overdose fatalities have changed over the past several years.

1. The table below gives the rate of illicit drug deaths in BC from 1991-2017. (source)
Year rate (in deaths per year per 100,000)
1991 3.5
1992 4.7
1993 9.9
1994 8.4
1995 5.7
1996 7.8
1997 7.6
1998 10.0
1999 6.8
2000 5.8
2001 5.8
2002 4.2
2003 4.6
2004 4.4
2005 5.5
2006 5.4
2007 4.7
2008 4.2
2009 4.6
2010 4.7
2011 6.5
2012 5.9
2013 7.3
2014 7.9
2015 11.1
2016 20.7
2017 31.6

We want to investigate how these rates are changing.

1. Calculate a line of best fit for the data, assuming a linear relationship, using a spreadsheet.
2. Suppose we have an equation of the form $R(t)=Ce^{rt}$, where $C$ and $r$ are constants. Show that the function $T(t)=\log(R(t))$ is a straight line, regardless of the base of our logarithm.
3. Take the logarithm of the rate data values in the table. We call the resulting data the ‘’transformed’’ data. Find a line of best fit for them as a function of time. Again, a spreadsheet is useful for this.
4. Use your line of best fit of the transformed data to approximate the data as a function of the form $R(t)=Ce^{rt}$. That is, what values of $C$ and $r$ does your fit suggest.
5. In both cases (linear fit to the original data and linear fit to the transformed data), interpret what the value of the slope of the fit indicates about the way the data is changing over time.
2. Below is a table giving the rate of overdose deaths from illicit drugs that involved fentanyl specifically (perhaps in conjunction with other drugs), 2012-2017. (source)
Year rate (in deaths per year per 100,000)
2011 and previous 0
2012 0.2
2013 1.1
2014 2.0
2015 3.2
2016 13.9
2017 (Jan-Aug) 25.6

Use this data to break down the data in the first table into fentanyl-related deaths and non-fentanyl-related deaths.

1. Fit the non-fentanyl related death rates since 2012 (calculate this by looking at both data sets provided) with a linear model. Interpret what the value of the slope of the fit indicates about the way the data is changing over time. Be sure to compare this slope to the slope found in part 1a.
2. Transform the fentanyl-related deaths data using the logarithm function (as carried out above) and fit the resulting data with a linear model. Ignore the pre-2012 data as they are all zero and cannot be transformed. Use your line of best fit of the transformed data to approximate the data as a function of the form $R(t)=Ce^{rt}$. That is, what values of $C$ and $r$ does your fit suggest. Interpret what the value of $r$ indicates about the way the data is changing over time. Be sure to compare this value of $r$ to the one found in part 1d.
3. Using the best-fit models for each data set (that is: the linear fit for the non-fentanyl overdose deaths from (a), and the exponential fit for the fentanyl-related deaths from (b)), predict the death rate expected in 2018 and 2020 from non-fentanyl related overdoses, and similarly for the fentanyl-related overdose death rate. Interpret what your analysis suggests about current trends in the overall rates of drug-related deaths.
4. Do you think your predictions are an accurate anticipation of the future? Why or why not?

Note: You are strongly encouraged to plot both the data and linear fits for all fits to data. These plots are not to be handed in. When a question ask for the best linear fit, you do not need to include the spreadsheets in your submission, just the resulting equation you get. Also, you can save yourself some effort by copying and pasting the data directly from the tables above into a spreadsheet. This should work smoothly in most spreadsheet apps.

3. If you or someone you know struggles with substance abuse, you can find resources here, here, and here.
4. This question will deal with the aftermath of the midterm. It is extremely important for you to identify and address your mistakes on the midterm before they hinder your further performance in the course. Download the solutions from the Midterm information page. For each question you got wrong, carefully go through it and characterize each, and every, mistake you made as:
A: Algebra or Arithmetic error
B: not knowing how to Begin the problem, not understanding what the question is asking for
C: not understanding a Concept or how to apply it in this particular context (specify symbolic, graphic, or geometric)
D: Distracted or careless mistake (rushed)
E: other reasons (please specify)
Create a list of errors you've made for every question. What types of errors are you prone to? What is your strategy to improve in these areas? You do not need to hand in a reworked and corrected version of your answers. You should just hand in a summary of your errors/issues and describe your strategy for avoiding them in the future.