# Chapter 10 Total and Partial Change

I do not feel obliged to believe that the same God who has endowed us with sense, reason, and intellect has intended us to forgo their use. —Galileo Galilei (1564-1642)

One of the most important ideas in science is experiment. In a simple, ideal form of an experiment, you cause one explanatory factor to vary, hold all the other conditions constant, and observe the response. A famous story of such an experiment involves Galileo Galilei (1564-1642) dropping balls of different masses but equal diameter from the Leaning Tower of Pisa.1 Would a heavy ball fall faster than a light ball, as theorized by Aristotle 2000 years previously? The quantity that Galileo varied was the weight of the ball, the quantity he observed was how fast the balls fell, the conditions he held constant were the height of the fall and the diameter of the balls. The experimental method of dropping balls side by side also holds constant the atmospheric conditions: temperature, humidity, wind, air density, etc.

Of course, Galileo had no control over the atmospheric conditions. By carrying out the experiment in a short period, while atmospheric conditions were steady, he effectively held them constant.

Today, Galileo’s experiment seems obvious. But not at the time. In the history of science, Galileo’s work was a landmark: he put observation at the fore, rather than the beliefs passed down from authority. Aristotle’s ancient theory, still considered authoritative in Galileo’s time, was that heavier objects fall faster.

The ideal of “holding all other conditions constant” is not always so simple as with dropping balls from a tower in steady weather. Consider an experiment to test the effect of a blood-pressure drug. Take two groups of people, give the people in one group the drug and give nothing to the other group. Observe how blood pressure changes in the two groups. The factor being caused to vary is whether or not a person gets the drug. But what is being held constant? Presumably the researcher took care to make the two groups as similar as possible: similar medical conditions and histories, similar weights, similar ages. But “similar” is not “constant.”

For non-experimentalists – people who study data collected through observation, without doing an experiment – a central question is whether there is a way to mimic “holding all other conditions constant.” For example, suppose you observe the academic performance of students, some taught in large classes and some in small classes, some taught by well-paid teachers and some taught by poorly-paid teachers, some coming from families with positive parental involvement and some not, and so on. Is there a way to analyze data so that you can separate the influences of these different factors, examining one factor while, through analysis if not through experiment, holding the others constant?

In this chapter you’ll see how models can be used to examine data as if some variables were being held constant. Perhaps the most important message of the chapter is that there is no point hiding your head in the sand; simply ignoring a variable is not at all the same thing as holding that variable constant. By including multiple variables in a model you make it possible to interpret that model in terms of holding the variables constant. But there is no methodological magic at work here. The results of modeling can be misleading if the model does not reflect the reality of what is happening in the system under study. Understanding how and when models can be used effectively, and when they can be misleading, will be a major theme of the remainder of the book.

## 10.1 Total and Partial Relationships

The common phrase “all other things being equal” is an important qualifier in describing relationships. To illustrate: A simple claim in economics is that a high price for a commodity reduces the demand. For example increasing the price of heating fuel will reduce demand as people turn down thermostats in order to save money. But the claim can be considered obvious only with the qualifier all other things being equal. For instance, the fuel price might have increased because winter weather has increased the demand for heating compared to summer. Thus, higher prices may be associated with higher demand. Unless you hold other variables constant – e.g., weather conditions – increased price may not in fact be associated with lower demand.

In fields such as economics, the Latin equivalent of “all other things being equal” is sometimes used: ceteris paribus. So, the economics claim would be, “higher prices are associated with lower demand, ceteris paribus.”

Although the phrase “all other things being equal” has a logical simplicity, it’s impractical to implement “all.” Instead of the blanket “all other things,” it’s helpful to be able to consider just “some other things” to be held constant, being explicit about what those things are. Other phrases along these lines are “taking into account …” and “controlling for ….” Such phrases apply when you want to examine the relationship between two variables, but there are additional variables that may be coming into play. The additional variables are called covariates or confounders .

A covariate is just an ordinary variable. The use of the word “covariate” rather than “variable” highlights the interest in holding this variable constant, to indicate that it’s not a variable of primary interest.

## 10.2 Example: Covariates and Death

This news report appeared in 2007:

Heart Surgery Drug Carries High Risk, Study Says.

A drug widely used to prevent excessive bleeding during heart surgery appears to raise the risk of dying in the five years afterward by nearly 50 percent, an international study found.

The researchers said replacing the drug – aprotinin, sold by Bayer under the brand name Trasylol – with other, cheaper drugs for a year would prevent 10,000 deaths worldwide over the next five years.

Bayer said in a statement that the findings are unreliable because Trasylol tends to be used in more complex operations, and the researchers’ statistical analysis did not fully account for the complexity of the surgery cases.

The study followed 3,876 patients who had heart bypass surgery at 62 medical centers in 16 nations. Researchers compared patients who received aprotinin to patients who got other drugs or no antibleeding drugs. Over five years, 20.8 percent of the aprotinin patients died, versus 12.7 percent of the patients who received no antibleeding drug. [This is a 64% increase in the death rate.]

When researchers adjusted for other factors, they found that patients who got Trasylol ran a 48 percent higher risk of dying in the five years afterward.

The other drugs, both cheaper generics, did not raise the risk of death significantly.

The study was not a randomized trial, meaning that it did not randomly assign patients to get aprotinin or not. In their analysis, the researchers took into account how sick patients were before surgery, but they acknowledged that some factors they did not account for may have contributed to the extra deaths. - Carla K. Johnson, Associated Press, 7 Feb. 2007

The report involves several variables. Of primary interest is the relationship between (1) the risk of dying after surgery and (2) the drug used to prevent excessive bleeding during surgery. Also potentially important are (3) the complexity of the surgical operation and (4) how sick the patients were before surgery. Bayer disputes the published results of the relationship between (1) and (2) holding (4) constant, saying that it’s also important to hold variable (3) constant.

In the aprotinin drug example, the total relationship involves a death rate of 20.8 percent of patients who got aprotinin, versus 12.7 percent for others. This implies an increase in the death rate by a factor of 1.64. When the researchers looked at a partial relationship (holding constant the patient sickness before the operation), the death rate was seen to increase by less: a factor of 1.48. In evaluating the drug, it’s best to examine its effects holding other factors constant. So, even though the data directly show a 64% increase in the death rate, 48% is a more meaningful number since it adjusts for covariates such as patient sickness. The difference between the two estimates reflect that sicker patients tended to be given aprotinin. As the last paragraph of the story indicates, however, the researchers did not take into account all covariates. Consequently, it’s hard to know whether the 48% number is a reliable guide for decision making.

The term partial relationship describes a relationship with one or more covariates being held constant. A useful thing to know in economics might be the partial relationship between fuel price and demand with weather conditions being held constant. Similarly, it’s a partial relationship when the article refers to the effect of the drug on patient outcome in those patients with a similar complexity of operation.

In contrast to a partial relationship where certain variables are being held constant, there is also a total relationship: how an explanatory variable is related to a response variable letting those other explanatory variables change as they will. (The corresponding Latin phrase is mutatis mutandis.)

Here’s an everyday illustration of the difference between partial and total relationships. I was once involved in a budget committee that recommended employee health benefits for the college at which I work. At the time, college employees who belonged to the college’s insurance plan received a generous subsidy for their health insurance costs. Employees who did not belong to the plan received no subsidy but were instead given a moderate monthly cash payment. After the stock-market crashed in year 2000, the college needed to cut budgets. As part of this, it was proposed to eliminate the cash payment to the employees who did not belong to the insurance plan. This proposal was supported by a claim that this would save money without reducing health benefits. I argued that this claim was about a partial relationship: how expenditures would change assuming that the number of people belonging to the insurance plan remained constant. I thought that this partial relationship was irrelevant; the loss of the cash payment would cause some employees, who currently received health benefits through their spouse’s health plan, to switch to the college’s health plan. Thus, the total relationship between the cash payment and expenditures might be the opposite of the partial relationship: the savings from the moderate cash payment would trigger a much larger expenditure by the college.

Perhaps it seems obvious that one should be concerned with the “big picture,” the total relationship between variables. If eliminating the cash payment increases expenditures overall, it makes no sense to focus exclusively on the narrow savings from the suspending the payment itself. On the other hand, in the aprotinin drug example, for understanding the impact of the drug itself it seems important to take into account how sick the various patients were and how complex the surgical operations. There’s no point ascribing damage to aprotinin that might instead be the result of complicated operations or the patient’s condition.

Whether you wish to study a partial or a total relationship is largely up to you and the context of your work. But certainly you need to know which relationship you are studying.

## Example: Used Car Prices

Figure 10.1 shows a scatter plot of the price of used Honda Accords versus the number of miles each car has been driven. The graph shows a pretty compelling relationship: the more miles that a car goes, the lower the price. This can be summarized by a simple linear model: price ~ mileage. Fitting such a model gives this model formula

price = 20770 - 0.10 × mileage.

Keeping in mind the units of the variables, the price of these Honda Accords typically falls by about 10 cents per mile driven. Think of that as the cost of the wear and tear of driving: depreciation.

As the cars are being driven, other things are happening to them. They are wearing out, they are being involved in minor accidents, and they are getting older. The relationship shown in Figure 10.1 takes none of these into account. As mileage changes, the other variables such as age are changing as they will: a total relationship.

In contrast to the total relationship, the partial relationship between price and mileage holding age constant tells you something different than the total relationship. The partial relationship would be relevant, for instance, if you were interested in the cost of driving a car. This cost includes gasoline, insurance, repairs, and depreciation of the car’s value. The car will age whether or not you drive it; the extra depreciation due to driving it will be indicated by the partial relationship between price and mileage holding age constant.

The most intuitive way to hold age constant is to look at the relationship between price and mileage for a subset of the cars; include only those cars of a given age. This is shown in Figure 10.2. The cars have been divided into age groups (less than 2 years old, between 3 and 4 years old, etc.) and the data for each group has been plotted separately together with the best fitting linear model for the cars in that group. From the figure it’s easy to see that the slope of the fitted models for each group is shallower than the slope fitted to all the cars combined. Instead of the price falling by about 10 cents per mile as it does for all the cars combined, within the 4-8 year old group the price decrease is only about 7 cents per mile, and only 3 cents per mile for cars older than 8 years.

By looking at the different age groups individually, you are holding age approximately constant in your model. The relationship you find in this way between price and mileage is a partial relationship. Of course, there are other variables that you didn’t hold constant. So, to be precise, you should describe the relationship you found as a partial relationship with respect to age.

## 10.3 Models and Partial Relationships

Models make it easy to estimate the partial relationship between a response variable and an explanatory variable, holding one or more covariates constant.

The first step is to fit a model using both the explanatory variable and the covariates that you want to hold constant. For example, to find the partial relationship between car price and miles driven, holding age constant, fit the model price ~ mileage+age. For the car-price data from Figure 10.1, this gives the model formula

price = 21330 - 0.077 × mileage - 538 × age

The second step is to interpret this model as a partial relationship between price and mileage holding age constant. A simple way to do this is to plug in some particular value for age, say 1 year. With this value plugged in, the formula for price as a function of mileage becomes

price = 21330 - 0.077 × mileage - 538 × 1 = 20792 - 0.077 × mileage

The partial relationship is that price goes down by 0.077 dollars per mile, holding age constant.

Note the use of the phrase “estimate the partial relationship” in the first paragraph of this section. The model you fit creates a representation of the system you are studying that incorporates both the variable of interest and the covariates in explaining the response values. In this mathematical representation, it’s easy to hold the covariates constant. If you don’t include the covariate in the model, you can’t hold it constant and so you can’t estimate the partial relationship between the response and the variable of interest while holding the covariate constant. But even when you do include the covariates in your model, there is a legitimate question of whether your model is a faithful reflection of reality; holding a covariate constant in a model is not the same thing as holding it constant in the real world. These issues, which revolve around the idea of the causal relationship between the covariate and the response, are discussed in Chapter 18.

## Aside: Partial change and partial derivatives

If you are familiar with calculus and partial derivatives, you may notice that this rate is the partial derivative of price with respect to mileage. Using partial derivatives allows one to interpret more complicated models relatively easily. For example, Figure 10.2 shows pretty clearly that the price vs mileage lines have different slopes for different age group. To capture this in your model, you might choose to include an interaction term between mileage and age. This gives a model with four terms:

price = 22140 - 0.094 × mileage - 750× age + 0.0034 × mileage × age

For this model, the partial relationship between price and mileage is not just the coefficient on mileage. Instead it is the partial derivative of price with respect to mileage, or:

price / ∂mileage = −0.094 + 0.0034 × age

Taking into account the units of the variables, this means that for a new car (age = 0), the price declines by $0.09/mile, that is, 9.4 cents per mile. But for a 10-year old car, the decline is less rapid: −0.094 + 10×0.0034 = −0.060 – only six cents a mile. ## 10.4 Adjustment The table contains the data on professional and sales employees of a large mid-western US trucking company: the annual earnings in 2007, sex, age, job title, how many years of employment with the company. Data such as these are sometimes used to establish whether or not employers discriminate on the basis of sex. sex earnings age title hiredyears M 35000 25 PROGRAMMER 0 F 36800 62 CLAIMS ADJUSTER 5 F 25000 34 RECRUITER 1 M 45000 44 CLAIMS ADJUSTER 0 M 40000 30 PROGRAMMER 5 A boxplot reveals a clear pattern: men are being paid more than women. (See Figure 10.3.) Fitting the model earnings ~ sex indicates the average difference in earnings between men and women: earnings = 35501 + 4735 sexM. Since earnings are in dollars per year, Men are being paid, on average,$4735 more per year than women. This difference reflects the total relationship between earnings and sex, letting other variables change as they will.

Notice from the boxplot that even within the male or female groups, there is considerable variability in annual earnings from person to person. Evidently, there is something other than sex that influences the wages.

An important question is whether you should be interested in the total relationship between earnings and sex, or the partial relationship, holding other variables constant. This is a difficult issue. Clearly there are some legitimate reasons to pay people differently, for example different levels of skill or experience or different job descriptions, but it’s always possible that these legitimate factors are being used to mask discrimination.

For the moment, take as a covariate something that can stand in as a proxy for experience: the employee’s age. Unlike job title, age is hardly something that can be manipulated to hide discrimination. Figure shows the employees’ earnings plotted against age. Also shown are the fitted model values of wages against age, fitted separately for men and women. It’s evident that for both men and women, earnings tend to increase with age. The model design imposes a straight line structure on the fitted model values. The formulas for the two lines are:

For Women Earnings $$=$$ 17178 + 530 age
For Men: Earnings $$=$$ 16735 + 609 age

From the graph, you can see the partial relationship between earnings and sex, holding age constant. Pick an age, say 30 years. At 30 years, according to the model, the difference in annual earnings is $1931, with men making more. At 40 years of age, the difference between the sexes is even more ($2722), at 20 years of age, the difference is less ($1140). All of these partial differences (holding age constant) are substantially less than the difference when age is not taken into account ($4735).

One way to summarize the differences in earnings between men and women is to answer this question: How would the earnings of the men have been different if the men were women? Of course you can’t know all the changes that might occur if the men were somehow magically transformed, but you can use the model to calculate the change assuming that all other variables except sex are held constant. This process is called adjustment.

To find the men’s wages adjusted as if they were women, take the age data for the men and plug them into the model formula for women. The difference between the earnings of men and women, adjusting for age, is $2125. This is much smaller than the difference,$4735, when earnings are not adjusted for age. Differences in age between the men and women in the data set appear to account for more than half of the overall earnings difference between men and women.

Of course, before you draw any conclusions, you need to know how precise these coefficients are. For instance, it’s a different story if the sex difference is 2125 ± 10 or if it is 2125 ± 5000. In the latter case, it would be sensible to conclude only that the data leave the matter of wage difference undecided. Later chapters in this book describe how to characterize the precision of an estimate.

Another key matter is that of causation. \$2125 indicates a difference, but doesn’t say completely where the difference comes from. By adjusting for age, the model disposes of the possibility that the earnings difference reflects differences in the typical ages of male and female workers. It remains to be found out whether the earnings difference might be due to different skill sets, discrimination, or other factors.

Sometimes the total relationship is the opposite of the partial relationship. This is Simpson’s paradox.

One of the most famous examples involves graduate admissions at the University of California in Berkeley. It was observed that graduate admission rates were lower for women than for men overall. This reflects the total relationship between admissions and sex. But, on a department-by-department basis, admissions rates for women were consistently as high or higher than for men. The partial relationship, taking into account the differences between departments, was utterly different from the total relationship.

## Example: Cancer Rates Increasing?

Consider another example of partial versus total relationships. In 1962, naturalist author Rachel Carson published Silent Spring (Carson 1962), a powerful indictment of the widespread use of pesticides such as DDT. Carson pointed out the links between DDT and dropping populations of birds such as the bald eagle. She also speculated that pesticides were the cause of a recent increase in the number of human cancer cases. The book’s publication was instrumental in the eventual banning of DDT.

The increase in deaths from cancer over time is a total relationship between cancer deaths and time. It’s relevant to consider a partial relationship between the number of cancer deaths and time, holding the population constant. This partial relationship can be indicated by a death rate: say, the number of cancer deaths per 100,000 people. It seems obvious that the covariate of population size ought to be held constant. But there are still other covariates to be held constant. The decades before Silent Spring had seen a strong decrease in deaths at young ages from other non-cancer diseases which now were under greater control. It had also seen a strong increase in smoking. When adjusting for these covariates, the death rate from cancer was actually falling, not increasing as Carson claimed.(Tierney 2007)

## 10.6 Explicitly Holding Covariates Constant

The distinction between explanatory variables and covariates is in the modeler’s mind. When it comes to fitting a model, both sorts of variables are considered on an equal basis when calculating the residuals and choosing the best fitting model to produce a model function. The way that you choose to interpret and analyze the model function is what determines whether you are examining partial change or total change.

The intuitive way to hold a covariate constant is to do just that. Experimentalists arrange their experimental conditions so that the covariates are the same. Think of Galileo using balls of the same diameter and varying only the mass. In a clinical trial of a new drug, perhaps you would test the drug only on women so that you don’t have to worry about the covariate sex.

When you are not doing an experiment but rather working with observational data, you can hold a covariate constant by throwing out data. Do you want to see the partial relationship between price and mileage while holding age constant? Then restrict your analysis to cars that are all the same age, say 3 years old. Want to know the relationship between breath-holding time and body size holding sex constant? Then study the relationship in just women or in just men.

Dividing data up into groups of similar cases, as in Chapter 4, is an intuitive way to study partial relationships. It can be effective, but it is not a very efficient way to use data.

The problem with dividing up data into groups is that the individual groups may not have many cases. For example, for the used cars shown in Figure 10.2 there are only a dozen or so cases in each of the groups. To get even this number of cases, the groups had to cover more than one year of age. For instance, the group labeled “age < 8” includes cars that are 5, 6, 7, and 8 years old. It would have been nice to be able to consider six-year old cars separately from seven-year old cars, but this would have left me with very few cases in either the six- or seven-year old group.

At the same time, it seems reasonable to think that 5- and 7-year old cars have something to say about 6-year old cars; you would expect the relationship between price and mileage to shift gradually with age. For instance, the relationship for 6-year old cars should be intermediate to the relationships for 5- and for 7-year old cars.

Modeling provides a powerful and efficient way to study partial relationships that does not require studying separate subsets of data. Just include multiple explanatory variables in the model. Whenever you fit a model with multiple explanatory variables, the model gives you information about the partial relationship between the response and each explanatory variable with respect to each of the other explanatory variables.

## Example: SAT Scores and School Spending

Chapter 7 showed some models relating school expenditures to SAT scores. The model sat ~ 1 + expend produced a negative coefficient on expend, suggesting that higher expenditures are associated with lower test scores. Including another variable, the fraction of students who take the SAT (variable frac) reversed this relationship.

The model sat ~ 1 + expend + frac attempts to capture how SAT scores depend both on expend and frac. In interpreting the model, you can look at how the SAT scores would change with expend while holding frac constant. That is, from the model formula, you can study the partial relationship between SAT and expend while holding frac constant.

The example also looked at a couple of other fiscally related variables: student-teacher ratio and teachers’ salary. The total relationship between each of the fiscal variables and SAT was negative – for instance, higher salaries were associated with lower SAT scores. But the partial relationship, holding frac constant, was the opposite: Simpson’s Paradox.

For a moment, take at face value the idea that higher teacher salaries and smaller class sizes are associated with better SAT scores as indicated by the following models:

sat = 988 + 2.18 salary - 2.78 frac

sat = 1119 - 3.73 ratio - 2.55 frac

In thinking about the impact of an intervention – changing teachers’ salaries or changing the student-teacher ratio – it’s important to think about what other things will be changing during the intervention. For example, one of the ways to make student-teacher ratios smaller is to hire more teachers. This is easier if salaries are held low. Similarly, salaries can be raised if fewer teachers are hired: increasing class size is one way to do this. So, salaries and student-teacher ratio are in conflict with each other.

If you want to anticipate what might be the effect of a change in teacher salary while holding student-teacher ratio constant, then you should include ratio in the model along with salary (and frac, whose dominating influence remains confounded with the other variables if it is left out of the model):

sat = 1058 + 2.55 salary - 4.64 ratio - 2.91 frac

Comparing this model to the previous ones gives some indication of the trade-off between salaries and student-teacher ratios. When ratio is included along with salary, the salary coefficient is somewhat bigger: 2.55 versus 2.18. This suggests that if salary is increased while holding constant the student-teacher ratio, salary has a stronger relationship with SAT scores than if salary is increased while allowing student-teacher ratio to vary in the way it usually does, accordingly.

Of course, you still need to have some way to determine whether the precision in the estimate of the coefficients is adequate to judge whether the detected difference in the salary coefficient is real – 2.18 in one model and 2.55 in the other. Such issues are introduced in Chapter 12.

## Aside: Divide and Be Conquered!

Efficiency starts to be a major issue when there are many covariates. Consider a study of the partial relationship between lung capacity and smoking, holding constant all these covariates: sex, body size, smoking status, age, physical fitness. There are two sexes and perhaps three or more levels of body size (e.g., small, medium, large). You might divide age into five different groups (e.g., pre-teens, teens, young adults, middle aged, elderly) and physical fitness into three levels (e.g., never exercise, sometimes, often). Taking the variables altogether, there are now 2 × 3 × 5 × 3 = 90 groups. It’s very inefficient to treat these 90 groups completely separately, as if none of the groups had anything to say about the others. A model of the form

lung capacity ~ body size + sex + smoking status + age + fitness

can not only do the job more efficiently, but avoids the need to divide quantitative variables such as body size or age into categories.

To illustrate, consider this news report:

Higher vitamin D intake has been associated with a significantly reduced risk of pancreatic cancer, according to a study released last week.

Researchers combined data from two prospective studies that included 46,771 men ages 40 to 75 and 75,427 women ages 38 to 65. They identified 365 cases of pancreatic cancer over 16 years.

Before their cancer was detected, subjects filled out dietary questionnaires, including information on vitamin supplements, and researchers calculated vitamin D intake. After statistically adjusting for [that is, holding constant] age, smoking, level of physical activity, intake of calcium and retinol and other factors, the association between vitamin D intake and reduced risk of pancreatic cancer was still significant.

Compared with people who consumed less than 150 units of vitamin D a day, those who consumed more than 600 units reduced their risk by 41 percent. - New York Times, 19 Sept. 2006, p. D6.

There are more than 125,000 cases in this study, but only 365 of them developed pancreatic cancer. If those 365 cases had been scattered around dozens or hundreds of groups and analyzed separately, there would be so little data in each group that no pattern would be discernible.

It’s tempting to think that including covariates in a model is a way to reach the truth: a model that describes how the real world works, a model that can correctly anticipate the consequences of interventions such as medical treatments or changes in policy, etc. This overstates the power of models.

A model design – the response variable and explanatory terms – is a statement of a hypothesis about how the world works. If this hypothesis happens to be right, then under ideal conditions the coefficients from the fitted model will approximate how the real world works. But if the hypothesis is wrong, for example if an important covariate has been left out, then the coefficients may not correctly describe how the world works.

In certain situations – the idealized experiment – researchers can create a world in which their modeling hypothesis is correct. In such situations there can be good reason to take the model results as indicating how the world works. For this reason, the results from studies based on experiments are generally taken as more reliable than results from non-experimental studies. But even when an experiment has been done, the situation may not be ideal; experimental subjects don’t always do what the experimenter tells them to and uncontrolled influences can sometimes remain at play.

It’s appropriate to show some humility about models and recognize that they can be no better than the assumptions that go into them. Useful object lessons are given by the episodes where conclusions from modeling (with careful adjustment for covariates) can be compared to experimental results. Some examples (from (Freedman 2008)):

• Does it help to use telephone canvassing to get out the vote? Models suggest it does, but experiments indicate otherwise.
• Is a diet rich in vitamins, fruits, vegetables and low in fat protective against cancer, heart disease or cognitive decline? Models suggest yes, but experiments generally do not.

The divergence between models and experiment suggests that an important covariate has been left out of the models.

## 10.8 The Geometry of Covariates and Adjustment

Figure 9.7 showed how the least-squares process for fitting a model like A ~ B + C projects a response variable A onto a model subspace B + C. Now consider the picture in that model subspace, after the projection has been done, as in Figure 10.5.

No residual vector appears in the picture because the perspective is looking straight down the vector, with the B + C model subspace drawn exactly on the plane of the page. (Recall that the residual vector is always perpendicular to the model subspace.)

In order to find the coefficients on B and C, you look for the number of steps you need to take in each direction, starting at the origin, to reach point A. In the picture, that’s about 2 B steps forward and 3 C steps backward, so the model formula will be A = 2 B - 3 C.

Now imaging that you had fit the model A ~ B without including the covariate C. In fitting this model, you would find the point on the B subspace that is as close as possible to A. That point is just about at the origin: 0 steps along B. So, including variable C as a covariate changes the model relationship between A and B.

Such changes in coefficients are inevitable whenever you add a new covariate to the model, unless the covariate happens to be exactly perpendicular – orthogonal – to the model vectors. The alignment of model vectors and covariates is called collinearity and will play a very important role in Chapter 12 in shaping not just the coefficients themselves but also the precision with coefficients can be estimated.

## Aside: Interaction terms and partial derivatives

Earlier, the idea of measuring effect size using a partial derivative was introduced. Partial derivatives also provide a way to think about interaction terms.

To measure the effect size of an explanatory variable, consider the partial derivative of the response variable with respect to the explanatory. Writing the response as z and the explanatory variables as x and y, the effect size of x corresponds to ∂z/∂x.

An interaction – how one explanatory variable modulates the effect of another on the response variable – corresponds to a mixed second-order partial derivative. For instance, the size of an interaction between x and y on response z corresponds to ∂²z / ∂x∂y.

A theorem in calculus shows that mixed partials have the same value regardless of the order in which the derivatives are taken. In other words, ∂²z / ∂x ∂y = ∂²z / ∂y∂x. This is the mathematical way of stating that the way x modulates the effect of y on z is the same thing as the way that y modulates the effect of x on z.

1. The picturesque story of balls dropped from the Tower of Pisa may not be true. Galileo did record experiments done by rolling balls down ramps.