2.8: Multiple Variables Weighing on Your mind?
- Page ID
- 148569
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
There are many types of multivariable models, but in this collaboration, we will concentrate on models which are similar to the linear models we studied previously, except that they have multiple input variables. In these multivariable models, each input variable has a coefficient, which is a number that multiplies the variable in the model. For example, if G = 3m − 2n, then both 3 and −2 are coefficients.
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- formulas using multiple variables are often used in real-world scenarios.
- multivariable models have one output variable and more than one input variable.
- a weighted average is a multivariable model which gives a better “average” for some sets of data.
- many problems are just variations of a weighted average problem, such as calculating grades and GPAs.
By the end of this collaboration, you should be able to
- interpret and use formulas with multiple variables.
- calculate the weighted average of a set of numbers in context.
- construct an expression for a weighted average problem.
- find missing information if the weighted average is known.
PROBLEM SITUATION 1: INVESTIGATING MODELS FOR RESTING ENERGY EXPENDITURE
In Preparation M.8, you read about the importance of calculating resting energy expenditure (REE). A variety of equations have been used to predict REE in critically ill patients.33 Some of the equations used to predict REE in critically ill patients are shown below.
Many of the equations below use multiple variables to predict REE. Mathematical models that use more than one input variable to produce an output variable are called multivariable models.
| American College of Chest Physicians equation | |
| REE = 25 * Weight |
| Harris-Benedict equation | |
|
Men: REE = 66.4730 + (13.7516 * Weight) + (5.0033 * Height) − (6.7550 * Age) |
Women: REE = 655.0955 + (9.5634 * Weight) + (1.8496 * Height) − (4.6756 * Age) |
| Penn State 1998 equation | |
| REE = (1.1 * value from Harris-Benedict) + (140 * Tmax) + (32 * VE) − 5,340 | |
| Penn State 2003 equation | |
| REE = (0.85 * value from Harris-Benedict) + (175 * Tmax) + (33 * VE) − 6,433 |
Notes about the variables in these equations:
- REE is measured in calories per day.
- Weight is measured in kilograms.
- Height is measured in centimeters.
- Age is in years.
- Tmax is the maximum body temperature in the past 24 hours, measured in Celsius.
- VE is a measure called minute volume, which is the volume of air that is inhaled and exhaled in one minute, measured in liters per minute. Normal levels are around 5–8 liters per minute.
(1) Which variables in the equations above have a positive (increasing) effect on REE? Take a minute to think about this on your own before sharing your ideas in your group. Explain how you determined your answer.
(2) Which variable in the equations above has a negative (decreasing) effect on REE? Take a minute to think about this on your own before sharing your ideas in your group. Explain how you determined your answer.
(3) Suppose Camila and Sofia, who are female twins, want to calculate their REE using the Harris-Benedict equation.
- Using the information in the table below, calculate the predicted REE for Camila and for Sofia. Do the calculations on your own before sharing your answers in your group.
| Camila | Sofia | |
| Weight (kg) | 61 | 62 |
| Height (cm) | 165 | 165 |
| Age (years) | 30 | 30 |
| Predicted REE |
(b) How much higher is Sofia’s predicted REE than Camila’s predicted REE? How does this relate to the coefficient for weight in the Harris-Benedict equation for females?
(c) Interpret the coefficient for Age in the Harris-Benedict equation for females.
(4) Imagine that you have just completed a moderate 45-minute workout and decide to measure your REE. Which of the equations might be used to support the notion that working out can raise your REE?
(5) Use the profile from Table 1 below to do the calculations (a) - (d). Round to the nearest calorie per day. Try each calculation on your own first before sharing your answers in your group. (Hint: Don’t forget that, in the equations, weight is measured in kilograms, height is measured in centimeters, and temperature is measured in degrees Celsius.)
Recall also that 1 lb ≈ 0.4536 kg, 1 inch = 2.54 cm, and a formula for converting degrees Fahrenheit (F) to degrees Celsius (C) is \(C = \frac{5}{9}(F − 32)\).
| Profile #1 | |
| Sex | Male |
| Age | 29 |
| Weight | 188 pounds |
| Height | 5’ 11” |
| Tmax | 98.6 °F |
| VE | 7 liters/minute |
| Profile #:_________ | |
| Formula | REE |
| American College of Chest Physicians | (a) |
| Harris-Benedict | (b) |
| Penn State 1998 | (c) |
| Penn State 2003 | (d) |
(6) How accurate do you think these formulas are?
PROBLEM SITUATION 2: TENNIS BALL VS CAR
Multivariable models can also have inputs with non-linear relationships to the output. An example is seen in a fundamental equation in physics for kinetic energy. Kinetic energy is the amount of energy an object has due to its motion. Think about an object moving toward you. The heavier the object and the faster it is moving, the more kinetic energy it has (the harder it will hit you). The equation for calculating kinetic energy is:
\[K = 0.5 * m * v^2\nonumber \]
where:
K = kinetic energy, in joules
m = mass of the object, in kilograms
v = speed of the object, in meters per second
(7) To consider the importance of kinetic energy, imagine being hit by a tennis ball that is traveling 50 miles per hour (it would probably hurt). Now imagine being hit by an average sized car moving at 65 miles per hour.
(a) To quantify the difference, calculate the kinetic energy of the tennis ball and the car. Try this on your own first before sharing your answers in your group. Round to the nearest one hundredth of a joule.
Hint: Tennis balls have a mass of 58.5 grams. An average sized car has a mass of 1500 kilograms.
Hint: Remember to convert units: 1 kilogram = 1000 grams; 1 mile per hour ≈ 0.447 meters per second.
(b) How many times greater is the car’s kinetic energy compared to the tennis ball’s kinetic energy? Round to the nearest whole number.
(c) If we instead compare the kinetic energy of two average sized cars, Car 1 moving at 50 miles per hour and Car 2 moving at 25 miles per hour, how many times greater is the kinetic energy of Car 1 compared to Car 2?
PROBLEM SITUATION 3: EVALUATING USED CARS
Your friend Gary is helping you search for a good used car to buy. He researched three cars yesterday and rated each car on a scale of 1 to 10 (10 is the best, 1 is the worst) in four different categories. Gary’s recommendations are shown in the table below. Gary recommended that you look at Car 3, because it has the highest average rating of the three cars.
| Car 1 | Car 2 | Car 3 | |
| Reliability | 8 | 6 | 7 |
| Gas Mileage | 6 | 7 | 5 |
| Interior Features/Comfort | 5 | 9 | 6 |
| Cargo space | 7 | 4 | 10 |
| Standard Average (Mean) | 6.5 | 6.5 | 7 |
(8) (a) What do you observe about how the three cars compare to each other across all categories?
(b) What is an explanation for why Car 3 has the highest overall average?
(9) What are the advantages and disadvantages of using a standard average to select the best overall car?
One way to show that some categories are more important than others would be to count that rating more times. For example, if Reliability is important, you can count it twice; if Comfort is even more important, you can count it four times. Your ratings might now look like this:
| Car 1 | Car 2 | Car 3 | |
| Reliability | 8 | 6 | 7 |
| Reliability | 8 | 6 | 7 |
| Gas Mileage | 6 | 7 | 5 |
| Interior Features/Comfort | 5 | 9 | 6 |
| Interior Features/Comfort | 5 | 9 | 6 |
| Interior Features/Comfort | 5 | 9 | 6 |
| Interior Features/Comfort | 5 | 9 | 6 |
| Cargo Space | 7 | 4 | 10 |
| Average (Mean) |
(10) Make a prediction for which car will have the highest overall rating and then calculate each average.
This kind of average, where some values are counted more times than others, is called a weighted average. The weights refer to how much each value is counted. In the above example, you could say that the categories of Gas Mileage and Cargo Space have weights equal to 1, Reliability has a weight of 2, and Interior Features/Comfort has a weight of 4.
So, the total number of weights is eight. (1 for Gas Mileage, 1 for Cargo Space, 2 for Reliability, 4 for Interior Features/Comfort = 8 total weights.)
Weighted average problems are specific cases of multivariable models. You might notice that the categories for rating the cars are acting as input variables used to calculate an overall score for “Car Quality” (weighted average), which is the output variable.
(11) Which of these following represents the multivariable model used to calculate the score for Car Quality (weighted average) in the previous problem? The following variables represent the individual rating in each category: R = Reliability; G = Gas Mileage; I = Interior Features/Comfort; C = Cargo Space.
(i) \(Car\;Quality = \dfrac{(R + G + I + C)}{4} \)
(ii) \(Car\;Quality = \dfrac{(R + G + I + C)}{8} \)
(iii) \(Car\;Quality = \dfrac{(2R + G + 4I + C)}{4} \)
(iv) \(Car\;Quality = \dfrac{(2R + G + 4I + C)}{8} \)
(12) Gary suggests using a different weighting strategy. He agrees with all of your preferences, but wants to use bigger weights with the following formula:
\[Car\;Quality = \dfrac{(10R + 5G + 20I + 5C)}{40} \nonumber \]
(a) What is the weight for the Interior Features/Comfort category?
(b) How many total weights are used in this equation?
(c) Using the same ratings in each category from above, find the weighted average for each car using this new equation. How do these weighted averages compare to the earlier weighted averages? What could explain your results?
(13) Imagine using this approach in choosing a car and create your own weighted average formula based on your own personal preferences. Use the same four categories: Reliability, Gas Mileage, Interior Features/Comfort, and Cargo Space, but choose relative weights for each category based on what you think is important in a vehicle.
- Write your weighted average formula.
(b) Explain how you chose the weights for the different categories.
(c) Use your formula to calculate the weighted average of each of the three cars.
(d) Which car has the highest weighted average according to your group's weighting? Which car would be the best out of the three for your group?
It was noted above that weighted average models are specific cases of multivariable models. There are many multivariable models that are not weighted average models, for example, the REE equations from Problem Situation 1.
(14) Consider the weighted average model from Question 12 and compare it to the Harris-Benedict equation for males’ REE (both shown below).
Weighted average: \(Car\;Quality = \dfrac{(10R + 5G + 20I + 5C)}{40} \nonumber \)
Harris-Benedict for males: REE = 66.4730 + (13.7516 * Weight) + (5.0033 * Height) − (6.7550 * Age)
(a) What differences do you notice between the equations?
(b) Why is the Harris-Benedict model for male REE not a weighted average?
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.
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33 http://rc.rcjournal.com/content/respcare/54/4/509.full.pdf