6 Statistical Parameters

The objectives of this chapter are to:

• Define what a parameter is in a statistical context

• See examples of such parameters

• Understand the concept of a parameter relation or a parameter equation

• Be able to set up parameter equations that isolate and directly pinpoint parameter differences in both the absolute and relative scales, using a regression equation framework.

• Do so before fitting any such (regression) equations to data, so that we can focus on the research objects without having data get in the way.

• See the unity (generality) in what we will be doing in the course, by seeing the big picture, i.e., the forests, not the trees.

6.1 Parameters

We begin by defining is meant by the term parameter in a statistical context

Definition 6.1 (Parameter) A constant (of unknown magnitude) in a (statistical) model. A numerical characteristic of a population, as distinguished from a statistic obtained by sampling.²⁵

The term can mean other things in other contexts. For example, in clinical medicine:

Any quantitative aspect/dimension of the client’s (patient’s) health, subject to measurement (by means of a test). (Example: systolic blood-pressure.)²⁶

6.1.1 The statistical parameters we will be concerned with

• \(\mu\) The mean level of a quantitative characteristic, e.g. the depth of the earth’s ocean or height of the land, or the height / BMI / blood pressure levels of a human population. [One could also think of mathematical and physical constants as parameters, even though their values are effectively ‘known.’ Examples where there is agreement to many many decimal places include the mathematical constant pi, the speed of light(c), and the gravitational constant G. The speed of sound depends on the medium it is travelling through, and the temperature of the medium. The freezing and boiling points of substances such as water and milk depend on altitude and barometric pressure]. At a lower level, we might be interested in personal characters, such as the size of a person’s vocabulary, or a person’s mean (or minimum, or typical) reaction time. The target could be a person’s ‘true score’ on some test – the value one would get if one (could, but not realistically) be tested on each of the (very large) number of test items in the test bank, or observed/measured continously over the period of interest.
      Later on we will address sitautions where the mean \(\mu\) is not the best ‘centre’ of a distribution, and why we might want to take some other feature, such as the median, or some other quantile, instead.

• \(\pi\) Prevalence or risk (proportion): e.g., proportion of the earth’s surface that is covered by water, or of a human population that has untreated hypertension, or lacks internet access, or will develop a new health condition over the next x years. At a lower level, we might be interested in personal proportions, such as what proportion of the calories a person consumes come from fat, or the proportion of the year 2020 the person spent on the internet, or indoors, or asleep, or sedentary.

• \(\lambda\) The speed with which events occur: e.g., earthquakes per earth-day, or heart attacks or traffic fatalities per (population)-year. At a lower level, we might be interested in personal intensities, such as the mean number of tweets/waking-hour a person issued during the year 2020, or the mean number of times per 100 hours of use a person’s laptop froze and needed to be re-booted.

Each of these three parameters refers to a characteristic of the overall domain, such as entire surface of the earth, or the entire ocean, or population. There are no indicators for distinguishing among subdomains, so they refer to locations / persons not otherwise specified. We will drill down later.

Especially for epidemiologic research, and also more generally, one can think of \(\pi\) and \(\lambda\) as parameters of occurrence (although the word occurrence usually has a time element, it can also be timeless: how frequently a word occurs in a static text, or a mineral in a rock). Prevalence is the proportion in a current state, and the 5-year risk is the expected proportion or probability of being in a new state 5 years from now. The parameter \(\lambda\) measures the speed with which the elements in question move from the original to the other state.

Even though the depths of the ocean, and blood pressures, are measured on a quantitative (rather than on all or none) scale, one can divide the scale into a finite number of bins/caterories, and speak of the prevalence (proportion) in each category. Conversely, one can use a set of descriptive parameters called quantiles, i.e, landmarks such that selected proportions, e.g., 0.05 or 5%, 25%, 50%, 75%, 95% of the distribution are to the left of (‘below’) these quantiles.

6.1.2 Occurence Parameters are not constants of nature

It has been noted in the philosophy of science that any science is concerned with functional relations of its objects (Friend and Feibleman, 1937). This proposition is quite evidently tenable for epidemiologic objects of research. Parameters of occurrence, such as the incidence rate for a particular illness, are not constants of nature. Rather, their magnitudes generally depend on — are functions of — a variety of characteristics of individuals — constitutional, behavioral, and/or environmental. Such relations, even if only remotely credible, are generally the objects of medical occurrence research. For example, one is quite usually interested in learning whether the rate of occurrence of some particular illness depends on (is related to or is a function of) gender — regardless of whether there is any express reason to surmise that it might be Miettinen.²⁷

Example 6.1 The prevalence of any given blood type based on the ABO antigen system,, while constant over gender and essentially constant over age, is not a constant of nature. It varies by ethnic groupings, for example. Thus the prevalence must be quantified in relation to—as a function of—ethnic group.

Example 6.2 For the occurrence of various values of blood pressure among people, one descriptive parameter is the median of the pressure. (This is a value such that the prevalence of its exceedance is 50%.) This parameter, again, is not a constant of nature but depends on age and other characteristics of individuals. For the quantitative nature of the age relation of systolic blood pressure, a rule of thumb used to be that it is, in mm Hg, 100 plus age in years." This rule expresses a regression model - a regression function - of the form P = A + B x Age. In this example, P, the occurrence parameter, is the median of systolic blood pressure, A = 100 mm Hg, and B = 1 mm Hg/yr.

Definition 6.2 (Determinants) The characteristics on which the magnitude of an occurrence parameter depends (causally or otherwise) are determinants of the parameter. Thus, in the examples given above, ethnic grouping is a determinant of prevalence of any given blood type, and age is a determinant of the median of systolic blood pressure.

Determinant has no implication as to causality in science — any more than in everyday locution: the current age of a person is “determined” by his/her year of birth (noncausally), just as the expected outcome of a disease is “determined” by the treatment that is used (causally). The relation of an occurrence measure to a determinant, or a set of determinants, is naturally termed an occurrence relation or an occurrence function. These relations are in general the objects of epidemiologic research.

6.1.3 Terminology

Before we go on, we need to adopt sensible terminology for referring generically to the states, traits, conditions or behaviours whose category-specific parameter values are being compared. We will use the term determinant.²⁸ It has several advantages over the many other terms used in different disciplines, such as exposure, agent, independent/explanatory variable, experimental condition, treatment, intervention, factor, risk factor, predictor.

The main advantage is that it is broader, and closer to causally neutral in its connotaion. Exposure has environmental connotations, and technically refers to an opportity to injest or mentally take on board a substance or message. Agent has causal connotations. The term independent variable suggests the investigator has control over it in a laboratory setting. The term explanatory is ambiguous as to the mechanism by which the parameter value in the index category got to be different from the value in the index category. Not all contrasts are experimentally formed. The term factor, and thus the term risk factor, are to be avoided because the word factor derives from the Latin facere, (the action of) doing, making, creating. Predictor makes one think of the future. The term regressor (or its shorthand, the ‘X’ ) won’t be understood by lay people.

While the word determine can suggest causality (e.g., demand determines the price), it also refers to fixing the form, position, or character of beforehand: two points determine a straight line; the area of a circle is determined by its radius.

There is considerable philosophical debate as to whether something causes something else. Some would argue that the extent to which genetics determines one’s personality is a causal concept. Others argue that since one cannot cannot consider the alternative, ones biological sex or age can not be considered a causal determinant or a risk factor (in the strict causal meaning of the word). They prefer to refer to them as risk indicators. We now move on to the parameter relations we will be concerned with, beginning with the simplest type.

6.2 Parameter Contrasts

In applied research, we are seldom interested in a single constant. Much more often we are interested in the contrast (difference) between the parameter values in different contexts/locations (Northern hemisphere vs Southern hemisphere), conditions/times (reaction times using the right versus left hand, or behaviour on weekdays versus weekends), or sub-domains or sub-populations (females vs males). Contrasts involving ‘persons, places, and times’ have a long history in epidemiology.

In this section, we will limit our attention to contrasts: a compariosn of the parameter values between 2 contexts/locations/sub-populations. Thus, the parameter function has just 2 possible ‘input’ values. The next section will address more general parameter functions.

6.2.1 Reference and Index categories

In many research contexts, the choice of ‘reference’ category (starting point, the category against which the other category is compared) will be obvious: it is the status quo (standard care, prevailing condition or usual situation, dominant hand, better known category). The ‘index’ category is the category one is curious about and wishes to learn more about, by contrasting its parameter value with the parameter value for the reference category.

In other contexts, it is less obvious which category should serve as the reference and the index categories, and the choice may be merely a matter of persepctive. If one is more famiar with the Northern hemisphere, it serves as a natural starting point (or ‘corner’ to use the terminology of Clayton and Hills,²⁹ or reference category). The choice of reference category in a longevity contrast between males and females, or in-hospital mortality rates or motor vehicle fatality rates during weekends versus weekdays, might depend on what mechanism one wishes to bring out. Or one might close as the reference category the one with the larger amount of experience, or maybe the one with the lower parameter value, so that the ‘index minus reference’ difference would be a positive quantity, or the ‘index: reference ratio’ exceeds 1.

6.2.2 Parameter relations in numbers and words

To make this concrete, we will use hypothetical (and very round) numbers and pretend we ‘know’ the true parameter values – in our example of the mean depth of the ocean in the Northern hemisphere (reference category) and Southern hemisphere (index category) – to be 3,600 metres (3.6Km) and 4,500 metres (4.5Km) respectively. Thus, the difference (South minus North) is 900 metres or 0.9Km.

If we wished to show the two parameter values graphically, we might do so using the format in Figure 6.1 panel (a), which shows the 2 hemisphere-specific parameter values – but forces the reader to calculate the difference.

Figure 6.1 panel (b) follows a more reader-friendy format, where the difference (the quantity of interest) is isolated: the original 2 parameters are converted to 2 new, more relevant ones.

Figure 6.1 panel (c) encodes the relation displayed in panel (b) in a single phrase that applies to both categories: Onto the ‘starting value’ of 3.8Km, one adds \(\Delta \mu\) = 0.9 Km only if the resulting parameter pertains to the Southern hemisphere. The 0.9 Km is toggled off/on as one moves from North to South.

Figure 6.1: Parameter relations in numbers and words

6.2.3 Parameter relations in symbols, and with the help of an index-category indicator

Panels (a) and (b) in Figure 6.2 repeat the information in panels (a) and (b) in Figure 6.1, but using Greek letters to symbolically represent the parameters. Just to keep the graphics uncluttered, the labels North and South are abbreviated to N and S and used as subscripts. Also, for brevity, the expression \[\Delta \mu =\mu_S - \mu_N.\]

The relation encoded in a single phrase shown in the Figure 6.1 panel (c) has a compact form suitable for verbal communication. The representation can be adapted to be more suitable for computer calculations. (The benefit of doing this will become obvious as soon as you try to learn the parameter values by fitting these models to actual data.) Depending on whether the hemisphere in question is the northern or southern hemisphere, the expression/statement ‘the specified hemisphere is the SOUTHERN hemisphere’ evaluates to a (logical) FALSE or TRUE. In the binary coding used in computers, it evaluates to 0 or 1, and we call such a 0/1 variable an ‘indicator’ variable.

We speak of INDICATOR variables, not DUMMY variables. The International Statistical Institute’s Dictionary of Statistical Terms objects to the name: the term is used, rather laxly, to denote an artificial variable expressing qualitative characteristics. The word DUMMY should be avoided. There is nothing dummy about an indicator variable.

Definition 6.3 (Indicator Variable) A variable with 0 and 1 as its only realizations, with realization 1 indicating something particular.

We encourage you to use, in your coding, meaningful variable names such as i.South or i.Southern (where i stands for indicator of) or i.Male. Don’t use the name sex or gender, where the coding is not self evident. If you think i.Male is over doing it, then use Male.

In Figure 6.2 panel (c), just to keep the graphics uncluttered, the name of the indicator variable SOUTHERN is abbreviated to S, and \(\mu_S\) is shorthand for the \(\mu\) cooresponding to whichever value (0 or 1) of \(S\) is specified (we could also write it as \(\mu | S\), or \(\mu\) ‘given’ \(S\)). Thus, the symbol \(\mu_0\) refers to the \(\mu\) when \(S=0\), or in longerhand, to \(\mu \ | \ S = 0\).

Figure 6.2: Parameter relations in symbols.

What does the equation in panel (c) remind you of?

Probably the equation of a line. In high school you may have learned it in the form \(A + B \times X\) that Miettinen used to describe the relation between median blood pressure and age.

Today, in statistics, these equations are referred to as regression equations, and the statistical model is called a regression model. The term regression is unfortunate, since it bears little relation to the original application. It concerned the phenomenon of reversion first described by Charles Darwin. Following his first studies of the sizes of the daughter seeds of sweet peas, his nephew Francis Galton, described the tendency:

Offspring did not tend to resemble their parent seeds in size, but to be always more mediocre (‘middling’, or closer to the mean) than they — to be smaller than the parents, if the parents were large; to be larger than the parents, if the parents were very small.³⁰

One of the first regression lines fitted to human data is Galton’s line depicting the rate of regression on hereditary stature where, using the term ‘deviate’ where today we would use ‘Z-scores.’

The Deviates of the Children are to those of their Mid-Parents as 2 to 3.³¹

Because he used Z scores (so the means in the parents and in the children were both 0) the equation of the line simplified to \[\mu(\textrm{Z-score in children of parents with mean z}) = 0 + (2/3) \times z\]

6.2.4 Don’t we need a cloud of points to have a regression line?

Although many courses and textbooks introduce regression concepts this way, the answer is NO. There is nothing in the regression formulation that specifies at which \(X\) values the mean \(Y\) values are to be determined. Unlike many textbboks that start with \(X\)s on a continuous scale, and then later have to deal with a 2-point (binary) \(X\), we are starting with this simplest case, and will move up later.

We are doing this for a few reasons: in epidemiology, the first and simplest contrasts involve just two categories, the reference category and the index category; a simple subtraction of 2 parameter values is easier to do and to explain to a lay person; and there is no argument about how the function behaves at the values between 0 and 1. There are no parameter values at Male = 0.4 or Male = 1.4, they are only at Male=0 and Male=1.

In addition, it is easier to learn the fundamental concepts and principles of regression if we can easily see what exactly is going on. Fewer blackbox formulae mean more transparency and understanding.Once we see how to represent parameter values in two determinant-categories, we can easily extend it to more than two, such as ethnic groups.

As we will see later on, when we have a value for a dental health parameter (eg the mean number of decayed, missing and filled DMF teeth) at \(X = 0\) parts per million of fluoride in the drinking water, and another parameter value at \(X = 1\) parts per million, we can only look at these 2 parameter values. If this is not enough, we would need to have (obtain) parameter values at the intermediate fluoride levels, or levels beyond 1 ppm, to trace out the full parameter relation, namely how the mean-DMF varies as a function of fluoride levels. If we have large numbers of observations at each level, then the DMF means will trace out a smooth curve. If data are limited, and the trace is jumpy/wobbly, we will probably resort to a sensible smooth function, the coefficients of which will have to be estimated from (fitted to) data.

This discussion leads on naturally to situations where the parameter varies over quantitative levels of a determinant - a topic considered in the next section.

But meantime, we need to answer this question: why limit ourselves to subtraction? why not consider the ratio of the two parameters, rather than their difference?

6.2.5 Relative differences (ratios) in numbers first

A ratio can be more helpful than a difference, especially if you are don’t have a sense of how large the parameter value is even in the reference category. As an example, on average, how many more red blood cells do men have than women? or how much faster are gamers’ reaction times compared with nongamers?

Recall our hypothetical mean ocean depths, 3.6 Km in the oceans in the Northern hemisphere (reference category) and 4.5Km in the oceans of the Southern hemisphere (index category). Thus, the S:N (South divided by North) ratio is 4.5/3.6 or 1.25.

Figure 6.3 panel (a) leaves it to the reader to calculate the ratio of the parameter values. In panel (b) the ratio (the quantity of interest) is isolated: again, the original 2 parameters are converted to 2 new, more relevant ones. Figure 6.3 panel (c) shows a single master-equation that applies to both hemispheres by togging off/on the ratio of 4.5/3.6.

Figure 6.3: Relative differences (ratios) in numbers first.

6.2.6 Relative differences (ratios) in symbols

To rewrite these numbers in a symbolic equation suitable for a computer, we again convert the logical ‘if South’ to a numerical Southern-hemisphere-indicator, using the binary variate \(S\) (short for Southern) that takes the value 0 if the Northern hemisphere, and 1 if the Southern hemisphere.

But go back to some long-forgotten mathematics from high school to be able to tell the computer to toggle the ratio off and on. Recall powers of numbers, where, for example, \(y\) to the power 2, or \(y^2\) is the square of \(y\). The two powers we exploit are 0 and 1. \(y\) to the power 1, or \(y^1\) is just \(y\) and \(y\) to the power 0, or \(y^0\) is 1.

We take advantage of these to write \[\mu_S = \mu \ | \ S \ = \mu_0 \ \times \ \Big\{ \frac{\mu_{South}}{\mu_{North}}\Big\}^S = \mu_0 \ \times Ratio \ ^ S.\]

You can check that it works for each hemisphere by setting \(S=0\) and \(S=1\) in turn. Thus, \[\log(y^S) = S \times \log(y)\]

Figure 6.4: Relative differences (ratios) in symbols.

Although this is a compact and direct way to express the parameter relation, it is not well suited for fitting these equations to data. However, in those same high school mathematics courses, you also learned about logarithms. For example, that \[\log(A \times B) = \log(A) + \log(B); \ \ \log(y^x) = x \times \log(y).\]

Thus, we can rewrite the equation in Figure 6.4 panel (c) as

\[\log(\mu_S) = \log(\mu \ | \ S) \ = \underbrace{\log(\mu_0)} \ + \underbrace{\log(Ratio)} \times \ S.\] This has the same linear in the two parameters form as the one for the parameter difference: the parameters are \(\underbrace{\log(\mu_0)}\) and \(\underbrace{\log(Ratio)}\) and they are made into the following ‘linear compound’ or ‘linear predictor’: \[\log(\mu_S) = \log(\mu \ | \ S) \ = \underbrace{\log(\mu_0)} \times \ 1 \ + \underbrace{\log(Ratio)} \times \ S.\]

The course is concerned with using regression software to fit/estimate these 2 parameters from \(n\) depth measurements indexed by \(S\).

6.3 Parameter functions

A very simple example of a function that describes how parameter values vary over quantitative levels of a determinant is the straight line shown in the upper right panel of the next figure. Here the determinant has the generic name X, and the equation is of the \(A + B \times X\) or \(B_0 + B_1 \times X\) or \(\beta_0 + \beta_1 \times X\) straight line form. Miettinen used the convention that the upper case letters \(A\) and \(B\) are used to denote the (true but unknown) coefficient values, whereas the lower case leters \(a\) and \(b\) are used to denote their empirical counterparts, sometimes called estimated coefficients or fitted coefficients. This sensible and simple convention also avoids the need, if one uses Greek letters for the theoretical coefficient values, to put ‘hats’ on them when we refer to their empirical counterparts, or ‘estimate/fit’ them. Fortunately, journals don’t usually allow investigators to use ‘beta-hats’; but this means that the investigators have to be more careful with their words and terms.

As we go left to right in Figure 6.5, the models become more complex. The simplest is the one of the left, in column 1, the one JH refers to as the mother of all regression models. It refers to a single or overall situation/population/domain, so \(X \equiv 1\), it takes on the value 1 in/for every instance/member. So the parameter equation is \(\mu_X = \mu \ | \ X \ = \mu \times 1.\) In column 2, there are 2 subdomains, indexed by the 2 values of the determinant (here generically called ‘X’), namely \(X = 0\) and \(X = 1.\) In the 3rd column, the number of of parameters is left unspecified, since the numbers of coefficients to specify a line/curve might vary from as few as 1 (if we were describing how the volume of a cube dependeds or, was is function of, its radius) to 2 (for a straight line that did not go through the origin, or for a symmetric S curve) to more than 2 (e.g., for a non-symmetric S curve, or a quadratic shape).

Figure 6.5: Grid of parameter contrasts that we aim to cover in this course.

A few more remarks on the panels in this Figure

• The 3 rows refer to the 3 core parameters we have given examples of above. All 3 are governed by the same principles, although there are more possibilities of different possible scales for some parameters.

• In column 1: there is just 1 parameter (value shown as a dot) corresponding to the overall population or the entire domain. You can think of it as the limiting or degenerate case of the columns to its right. One can still write it in a regression model.

A one parameter model is sometimes referred to as a null or incercept only regression model. We will exploit this idea to take a more holistic/general and economical approach to this introductory course. Many textbooks/courses do not mention regression models until quite late, and spend a lot of time on 1-sample (and even 2-sample) problems without pointing out that these are merely sub-cases of regression models. This silos practice of promoting/learning a separate software routine for dealing with a 1 sample problem, when one can get the same answer from a regression routine, leads to dead ends and wastes time.
Once we get to fitting/estimating a mean (or proportion or rate) parameter to/from data, we will encourage doing so within a regression framework.

• In setting (column) 2, there are 2 parameter values, one for the reference category and one for the index category of the determinant. As we have seen, how they relate to each other can be can be expressed in a number of different ways. A common and useful way is via a parameter equation that contains a parameter for the reference category and a comparative parameter (some measure of the difference between the two parameters) – the latter is often of most interest.

• In setting (column) 3, the parameter equation traces the parameter over a continuum of possible values of the determinant, using as many coefficients as are needed. In this particular diagram, the values of the determinant (X) are shown starting at X = 0, but this does not have to be. In data analysis, one often shifts the X origin, so that the ‘intercept’ makes more sense. For example, if one was plotting world temperatures, or ice-melting dates (see Chapter on Computning) against calendar year, it would be better to have the incercept refer to the fitted temperature for when the series begins, rather than when our current Western calendat begins (at the year 0 AD). Likewise, if we were describing the relation between ideal weight and height it is good to start near where people’s heights are. Thus, ‘100 pounds for a height of 5 feet, with five additional pounds for each added inch of height’ for women, and ‘106 pounds for a height of 5 feet, and six additional pounds for every added inch of height.’ for men. Of course, if you wish, for women you could use the mathematically equivalent ‘-300 pounds for a height of 0 feet, with five additional pounds for each added inch of height’ but it is not that easy to remember, and doesn’t apply for much of the (unspecified) height range!

A few remarks on associated terminology

Instead of regression models, some textbooks and courses refer to linear models :

Definition 6.4 (Linear model) Formulation of the mean/expectation of (the distribution of) a random variate (Y) as a linear compound of a set \(B_0 , B_1 , B_2 , \dots\) of parameters: as \(B_0 + B_1 X_1 + B_2 X_2 + \dots\).³²

The meaning of ‘linear’ in the appellation of this model has nothing to do with straight lines; it refers to the mathematical concept of linear compound: given quantities Q\(_1\), Q\(_2\), etc., a linear compound of these is the sum C\(_1\)Q\(_1\) + C\(_2\)Q\(_2\) + …, where C\(_1\) etc. are the ‘coefficients’ that define a particular linear compound of the set of quantities constituted by the Qs.

So, the ‘general linear model’ is linear in the sense that the dependent parameter, M, is formulated as a linear compound of the independent parameters B0, B1, etc., the coefficients in this linear compound being 1, X1, etc. The model is, in this way, ‘linear in the parameters.’

6.4 Analysis of Variance

Statistics courses in the social sciences, the biological laboratory sciences, and other experimentally-based sciences, typically move on from 1- and 2-sample procedures (unfortunately, mainly focusing on statistical tests) to ‘analysis of variance’ models

Miettinen explains an ’analysis of variance models this way:³³

In the ‘analysis of variance model,’ the random variate at issue – Gaussian – has a mean whose value depends on a nominal-scale determinant, a nominal scale being characterized by discrete categories without any natural order among them. The names of the (nominal) categories, some N in number, could be Category 1, Category 2, … , Category N. The term for the model is a misnomer. For, at issue is not analysis but synthesis of data, and the synthesis is not directed to learning about the variance of the random variate; it focuses on the mean, the relation of the mean to the (nominal-scale) determinant of it.

A simple example of these models might address the mean of systemic blood-pressure – defined as the weighted average of the diastolic and systolic pressures with weights 2/3 and 1/3, respectively – in relation to ethnicity, represented by three categories. An ‘analysis-of- variance’ model would define a random variate (Y) as representing the numerical value of the pressure (statistical variates inherently being numerical) and having a Gaussian distribution with means M1, M2, and M3 in those ethnicity categories 1, 2, and 3, respectively, with the variance of the distribution invariant among them. The random variate (Y) is the ‘dependent’ variate in the meaning that the value of its mean depends on ethnicity; and the ethnicity categories are represented in terms of suitably-defined ‘independent’ – non-random – variates (Xs).

The form of the ‘analysis-of-variance’ model in this simple example is: \(M = B_0 + B_1 X_1 + B_2 X_2\), where \(M\) is the mean of \(Y\) and the two independent variates are indicators of two particular ones of the three ethnic categories. One possibility in this framework is to take \(X_1\) and \(X_2\) to be indicators of Category 2 and Category 3, respectively – an indicator variate being one that takes on the value 1 for the category it indicates, 0 otherwise.

In terms of this model, \(B_0\) is the value of \(M\) when \(X_1 = X_2 = 0\), that is, for Category 1 (i.e., \(B_0 = M_1\)); and for Category 2 and Category 3 the values of \(M\) are represented by \(B_0 + B_1\) and \(B_0 + B_2\), respectively (i.e., \(M_2 = B_0 + B_1\), and \(M_3 = B_0 + B_2\)). Thus, the difference between \(M_1\) and \(M_2\) is represented by \(B_1\); \(B_2\) represents the difference between \(M_1\) and \(M_3\); and the difference between \(M_2\) and \(M_3\) is the difference between \(B_1\) and \(B_2\).

In this ‘analysis-of-variance’ framework it is feasible to accommodate, jointly, whatever number of nominal-scale determinants of the magnitude of the mean of the dependent variate. A simple example of this is the addition of the two categories of gender for consideration jointly with the three categories of ethnicity. These two determinants jointly imply a single nominal-scale determinant with six categories (as each of the three categories of ethnicity is split into two subcategories based on gender).

When involved in the definition of the independent variates is only a single determinant, the model is said to be for ‘one-way analysis of variance’; with two determinants the corresponding term (naturally) is ‘two-way analysis of variance’; etc.

6.5 Phraseology to avoid

It is quite common to hear a regression coefficient (fitted or theoretical) interpreted this way:

“For every 1 unit increase in X, the ‘Y’ parameter increases by \(\beta_X\) units.”

or as follows

“As (when) you increase X by 1 unit, you increase the Y parameter by \(\beta_X\) units.”

We pick up this terminology very early, maybe even back in high school, and from other people around us. But, in interpreting the B = 1 mm Hg/yr in Miettenen’s example (100 plus age in years), should we use such phrases?

Or, since you don’t know the source of, or the data behind this rule, you can take a look at the distributions of some anthropometric chacacteristics (height, weight, forced expiratory volume, FEV) measured cross-sectionally, in different populations – Busselton, Australia and rural Southwest Ethiopia – in 1972 and 1992. By eye, try to estimate the slope you would get if you regressed the age-and sex-specific means or medians on the ages. and then summarize the gradient across age.

Remember that these these subjecsts aren’t aging or going anywhere, and nobody was watching them age.

It is more accurate to say:

People who were aged a+1 years at the time of the survey had heights/weights/FEVs that were t.tt units higher/lower than people who were aged a years.

or

The mortality rate was u.u units higher/lower (or u.u times higher/lower) in the experience in the index category than the reference category.

This way, you are telling the reader that this is a static source, and not a dynamic situation where conditions are being manipulated by the investigators, or the subjects being watched as their ages go up [for many readers, the word ‘increased’ implies that some human force deliberately changed the dial, and turned the X up or down, as one could do with temperature or humidity in a laboratory.]

One of JH’s favourite examples of people being misled into thinking that a cross-sectional dataset allows you to say that ‘as people get older, they …’ is the McGill epidemiology department’s studies, in the 1960s, on the health of the more than 10,000 millers and miners of asbestos. These workers were born between 1890 and 1920. In cross-sectional studies, there were gradients in mean height across attained age. It would be easy to give them a ‘as people get older, they shrink in height’ or they ‘gain in height’ interpretation. It is easy to overlook the fact that some of these were children and adolescents during the Depression.

6.6 Exercises

So far, we have only dealt with equations involving a difference and the ratio of two \(\mu\) parameters.

1. Extend the graphs and the equations (for the difference of means and the ratio of means) to the \(\pi\) parameter. Use as an example the proportions of the surfaces of the Northern (reference category) and Southern hemisphere (index category) covered by water, i.e, \(\pi_{North}\) and \(\pi_{South}.\) Use the hypothetical values \(\pi_{North} = 0.65\) and \(\pi_{South} = 0.75.\)³⁴

2. Instead of focusing on the proportions covered by water, focus on the proportions covered by land. How does the difference of the two proportions relate to the difference calculated in 1?³⁵

3. How does the ratio of the two proportions relate to the ratio calculated in Q2? i.e., is one the reciprocal of the other?³⁶

4. Can you think of different scale, where the ratio when the focus is land IS just the reciprocal of the ratio when the focus is water?³⁷

5. If you can, show that the log of the ratio when the focus is land IS just the negative of the log of the ratio when the focus is water?³⁸

6. Extend the graphs and the equations (for the difference of means and the ratio of means) to the \(\lambda\) parameter. Use as an example the mean number of earthquakes per year in the Northern (reference category) and Southern hemisphere (index category) , i.e, \(\lambda_{North}\) and \(\lambda_{South}.\) Use the hypothetical values \(\lambda_{North} = 5.0\) and \(\lambda_{South} = 7.5\)³⁹