Pairwise Regression Weight Contrasts: Models for Allocating Psychological Resources

Abstract

Researchers examine contrasts between analysis of variance (ANOVA) effects but seldom contrasts between regression coefficients even though such coefficients are an ANOVA generalization. Regression weight contrasts can be analyzed by reparameterizing the linear model. Two pairwise contrast models are developed for the study of qualitative differences among predictors. One leads to tests of null hypotheses that the regression weight for a reference predictor equals each of the other weights. The second involves ordered predictors and null hypotheses that the weight for a predictor equals that for the variables just above or below in the ordering. As illustration, qualitative differences in high school math course content are related to math achievement. The models facilitate the study of qualitative differences among predictors and the allocation of resources. They also readily generalize to moderated, hierarchical, and generalized linear forms.

Keywords

linear regression generalized linear regression randomized experiments hierarchical linear regression mathematics coursework mathematics achievement ANOVA

Within the analysis of variance (ANOVA) context, hypotheses about contrasts between effects are common. While ANOVA is a special case of the general linear model where the regression weights correspond to effects, there is a smaller literature on testing hypotheses about contrasts between regression weights in a single group (Buse, 1982; Carlton-Ford, 1993; Davison et al., 2023; Davison et al., 2020; Engle, 1984; Evans & Savin, 1982; Graybill, 1976; Kiviet, 1986; Rindskopf, 1984). This literature has at least two camps. The first involves a penalty function (Buse, 1982; Engle, 1984; Evans & Savin, 1982; Kiviet, 1986). This approach adopts maximum likelihood optimization and χ² test statistics. The second approach, called the substitution approach by Davison et al. (2020), uses ordinary least squares. It is named substitution because it uses a reparameterized linear model that replaces predictors by proxies that are linear combinations of the original predictors. Weights in the reparameterized model correspond to contrasts between the original weights. While first developed for linear regression (Carlton-Ford, 1993; Davison et al., 2020; Rindskopf, 1984), this approach was extended to generalized linear models for categorical criteria (Davison, Jew, et al., 2014) and latent variable modeling (Davison, Chang, et al., 2014). Here, we give two reparameterizations of the model, in which parameters are contrasts between regression weights. We also do a simulation study comparing Type I error and power for the Lagrange multiplier (LM) penalty and substitution methods.

One of our interests is testing the hypothesis of equality for a linear weight contrast of two predictor variables $H_{0} : b_{v} = b_{v'}$ . Under strict assumptions, this test is equivalent to the test of the correlation of the two variables with the criterion Y: $H_{0} : ρ_{v . Y} = ρ_{v' . Y}$ . Our interest is not in using the null hypothesis $H_{0} : b_{v} = b_{v'}$ as a proxy for $H_{0} : ρ_{v . Y} = ρ_{v' . Y}$ . We are interested in testing $H_{0} : b_{v} = b_{v'}$ as a way of answering the question of whether adding a point to predictor X_v has the same effect on the expected value of the criterion as adding a point to $X_{v'}$ :

E (Y | X_{1}, X_{2}, \dots X_{v} + 1, X_{v^{'}}, \dots, X_{V}) - E (Y | X_{1}, X_{2}, \dots X_{v}, X_{v'} + 1, \dots ., X_{V}) = 0.

In sum, this is akin to asking whether predictors differ qualitatively relative to the criterion.

What we propose will likely be of special interest when the predictors are in common units. For instance, calories come in carbohydrates, fats, and proteins. In a study of weight loss, one may be interested in whether a reduction of one calorie is associated with the same expected weight loss regardless of the source of the calorie. Cooper et al. (1992) studied whether bone loss from caffeine varied with the source of caffeine: coffee, tea, or other. Stated as a regression problem, when bone loss is regressed onto coffee caffeine, tea caffeine, and other caffeine; does one unit increase of caffeine lead to the same expected bone loss irrespective of its source? Below, we regressed high school math achievement test scores onto the number of instructional units taken by students in ten high school math subject areas (e.g., algebra 1, trigonometry, statistics). The research question is whether a unit of math instruction is associated with the same expected change in math achievement irrespective of subject matter? The question of whether the effect of a unit change in predictor is equivalent for the criterion is of special interest when the predictors are measured in the same unit.

Our goal is to extend the substitution approach for pairwise contrasts with more than two predictors. These models can be of interest for allocating psychological resources. One may wish to allocate resources to get more “bang for the buck,” which suggests qualitative differences between the resources relative to a criterion. The models we present also have inherent theoretical meaning. It is easier to explain the reparameterized model as one of resource allocation after deriving the model. Thus, we first explain the model.

The reparameterized model can be extended to a moderated model for testing interactions between the contrasts and person variables, such as sex or age. It can also be used to study the effect on the variance accounted for by deleting predictors. Deleting a predictor X _v from the ordinary model will generally have a different effect on R ² than will deleting that same variable from the reparameterized model obtained through substitution. The model also readily extends to generalized linear and hierarchical forms. It can be used with categorical dependent variables to test hypotheses about regression weight contrasts. None of this requires specialized software.

Substitution Approach

Substitution involves replacing predictors in the usual regression model with proxies, so weights on some predictors in the new model are contrasts between the pairs of weights in the original model. Each proxy variable is a linear combination of the original predictors. Rindkopf (1984) showed how to reparameterize the usual regression model, so the resulting parameters satisfy specified, linear constraints. Carlton-Ford (1993) used Rindkopf’s idea to reparameterize the original model for two predictors to test whether the two regression weights are equal. There are many ways to extend Carlton-Ford’s approach to more variables and we present two. The first we call the referent approach where one predictor in the original model (e.g., X₁ ) is the referent, and the others are focal variables, predictors 2,…, V. The referent could be a standard or control condition. This formulation leads to regression weights in the reparameterized model that are contrasts between weights in the original model b _v − b ₁ for all focal variables v >1. For the second approach, ordered predictor approach, weights in the reparameterized model are contrasts b _v − b _v−1 from the original model. This formulation presumes an ordering of predictors. The ordering may be from theory, as in our following example. There are infinite reparameterizations possible with our two being potentially theoretically meaningful.

Here, we describe the substitution approach for four predictors and extensions to more. We show the referent and then the ordered approach. For both, it is posited that qualitative differences between predictors moderate the size of the predictor weights. For instance, if the predictors are calories, we posit that the expected change in the criterion resulting from increased calories varies depending on the source of the calorie. Thus, qualitative differences in carbohydrates, fats, and proteins moderate the expected change in weight. An important aspect of our approach is that it allows one to quantify effects associated with qualitative features of predictors.

Referent Substitution Approach

The following equations are in scalar form, as will be our proofs

Y = b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + \dots + b_{V} X_{V} + a + e,

where Y is the criterion, b_v is the regression weight for predictor v, X_v is the predictor v, a is the intercept, and e is the error. Let X ₁ be the reference variable, and the remaining predictors be focal variables. To reparameterize Equation 2a for the referent approach, we add and subtract $\sum_{v = 2}^{v = V} b_{1} X_{v}$ to the right side of Equation 2a. In the case of four predictors, we get

Y = b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} + a + \sum_{v = 2}^{v = 4} b_{1} X_{v} - \sum_{v = 2}^{v = 4} b_{1} X_{v} + e,

= (b_{1} X_{1} + \sum_{v = 2}^{v = 4} b_{1} X_{v}) + (b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} - \sum_{v = 2}^{v = 4} b_{1} X_{v}) + a + e,

= b_{1} (X_{1 +} \sum_{v = 2}^{v = 4} X_{v}) + (b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} - b_{1} X_{2} - b_{1} X_{3} - b_{1} X_{4}) + a + e,

If we let T ₁ be the sum of all predictor variables (1 to V), the first term on the right of Equation 2d becomes $b_{1} T_{1} = b_{1} \sum_{v = 1}^{v = 4} X_{v}$ = $b_{1} (X_{1} + X_{2} + X_{3} + X_{4})$ , and we can rewrite Equation 2d as

Y = b_{1} T_{1} + (b_{2} X_{2} - b_{1} X_{2}) + (b_{3} X_{3} - b_{1} X_{3}) + (b_{4} X_{4} - b_{1} X_{4}) + a + e,

= b_{1} T_{1} + (b_{2} - b_{1}) X_{2} + (b_{3} - b_{1}) X_{3} + (b_{4} - b_{1}) X_{4} + a + e .

Fitting Equation 2f involves regressing Y onto $T_{1}, X_{2}, X_{3}, and X_{4}$ . In this substitution, only X ₁ in the original model is replaced by a proxy sum T ₁. The weight on X ₁ in Equation 2a equals that for T ₁ in Equation 2f. For each of the focal predictors, the new weight is a contrast between the original weight for that predictor and the original weight for the reference predictor (from Equation 2a): $b_{2} - b_{1}$ , $b_{3} - b_{1}$ , and $b_{4} - b_{1}$ . The familiar t-statistic for testing the null hypothesis that the coefficient equals 0.0 now tests whether each contrast equals 0. For any number of predictors, Equation 2f generalizes to

Y = b_{1} T_{1} + \sum_{v = 2}^{v = V} (b_{v} - b_{1}) X_{v} + a + e .

Referent substitution is similar to reference cell coding in ANOVA. If for four groups, we let Group 1 be the referent and then let: X₂ = 1 when Group = 2 and 0 otherwise, X₃ = 1 when Group = 3 and 0 otherwise, and X₄ = 1 when Group = 4 and 0 otherwise. Using Equation 2a, the predictive model is: $Y = a + b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} + e$ . Given that these models consist of constants where the results are the Group means, the effective models become

Y = a + e If Group = 1 a = \bar{Y_{1}},

Y = a + b_{2} + e If Group = 2 b_{2} = \bar{Y_{2}} - \bar{Y_{1}},

Y = a + b_{3} + e If Group = 3 b_{3} = \bar{Y_{3}} - \bar{Y_{1}}, and

Y = a + b_{4} + e If Group = 4 b_{4} = \bar{Y_{4}} - \bar{Y_{1}} .

The weights b ₂, b ₃, and b ₄ are mean contrasts between each group and Group 1. These weights are significant if the focal group’s mean differs from the referent group’s mean.

Ordered Substitution Approach

The Appendix shows that Equation 2a for four predictors can also be reparameterized as

Y = b_{1} \sum_{v = 1}^{v = 4} X_{v} + (b_{2} - b_{1}) \sum_{v = 2}^{v = 4} X_{v} + (b_{3} - b_{2}) \sum_{v = 3}^{v = 4} X_{v} + (b_{4} - b_{3}) X_{4} + a + e,

= b_{1} T_{1} + (b_{2} - b_{1}) T_{2} + (b_{3} - b_{2}) T_{3} + (b_{4} - b_{3}) X_{4} + a + e .

Generalized to any number of predictors V, Equation 4b becomes

Y = b_{1} T_{1} + \sum_{v = 2}^{v = V - 1} (b_{v} - b_{v - 1}) T_{v} + (b_{V} - b_{V - 1}) X_{V} + a + e .

In Equation 4a–c, the v*th predictor (v* = 1,…, V − 1) is a sum $T_{v *}$ over predictors v* to V: $T_{v *} = \sum_{v = v *}^{v = V} X_{v}$ . The derivation of Equation 4b in the Appendix involves three steps to replace the three predictors, X ₁, X ₂, and X ₃ with corresponding sums, T ₁, T ₂, and T ₃. Each step involves adding and subtracting a sum to the result of the previous step. If there are V predictors, ordered substitution involves replacing the first V − 1 predictors with V − 1 sums, and the proof involves V − 1 steps. Equation 4c can generalize to 4d, since $T_{v} = \sum_{v = V}^{v = V} X_{v} = X_{V}$ :

Y = b_{1} T_{1} + \sum_{v = 2}^{v = V} (b_{v} - b_{v - 1}) T_{v} + a + e .

As with referent substitution, the ordered formulation also has a representation in ANOVA. For four groups let X₁ = 1 when Group > 2 (i.e., Group = 2, 3, or 4) and 0 otherwise, X₂ = 1 when Group > 3 (i.e., Group = 3 or 4) and 0 otherwise, X₃ = 1 when Group > 4 (i.e., Group = 4) and 0 otherwise. Given the model: $Y = a + b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + e$ , the effective models are

Y = a + e If Group = 1, a = \bar{Y_{1}},

Y = a + b_{1} + e If Group = 2, b_{1} is a contrast for Groups 1 and 2, \bar{Y_{2}} - \bar{Y_{1}},

Y = a + b_{1} + b_{2} + e If Group = 3, b_{2} is a contrast for Groups 2 and 3, \bar{Y_{3}} - \bar{Y_{2}},

Y = a + b_{1} + b_{2} + b_{3} + e If Group = 4, b_{3} is a contrast for Groups 3 and 4, \bar{Y_{4}} - \bar{Y_{3}} .

Two aspects of Equation 4a–c are of interest. First, for the last three predictors, the regression weight is a contrast between two regression weights in Equation 2a. The t-statistic for testing a regression weight equals zero now tests whether contrasts between adjacent regression weights in Equation 2a are equal. Testing the null hypothesis that the X ₂ weight is zero gives a test of the null hypothesis H ₀: b ₂−b ₁= 0, for the first two regression weights in Equation 2a. In our math achievement example, we are interested in whether the predictors form a hierarchy in that the weight for each successive predictor is larger than the previous: b _v > b _v-1 for all v > 1.

A second important feature of Equation 4c (and Equation 2f) is its submodel that includes only the first predictor in the model:

Y = b \sum_{v = 1}^{v = V} X_{v} + a + e = b T_{1} + a + e .

This submodel is nested within the model of Equation 2a. It is a model where all predictors have the same regression weight: $b_{1} = b_{2} = b_{3} = b_{4} = b$ . This leads to a test of the global null hypothesis $H_{0} :$ $b_{1} = b_{2} = b_{3} = b_{4} = b$ using the ordinary F-statistic to test the null hypothesis that the variation accounted for by the full model of Equation 2a (or any reparameterizations) is the same as the variance accounted for by the nested model of Equation 6:

F_{(V - 1, N - V - 1)} = \frac{(S S E_{R} - S S E_{F}) / (N - V - 1)}{S S E_{F} / (V - 1)},

where $S S E_{R} and S S E_{F}$ are the sum of squares residual for the restricted (Equation 6) and full (Equation 2a) models, respectively.

Equation 4b and c gives rise to multicollinearity concerns. Because the predictor sums in these equations contain overlapping sums of variables, those predictor sums may be highly correlated. A simulation study reported in the following gives an indication of how this multicollinearity affects the global hypothesis test and the hypothesis tests for regression weight contrasts.

Resource Allocation Model

For Equation 2a, the regression weight for X_v is the expected change in Y given a one-unit increase in X_v holding all other predictors constant. As a model of behavior, this assumption may be violated in some contexts. Consider a study in which the units of study are organizations, the criterion is a measure of organizational success, and the predictors are budget allocations to various categories: for example, personnel, travel, equipment, and so on. Once the manager of an organization receives the annual budget, the manager can allocate dollars across categories but the total is fixed. The manager cannot allocate an additional dollar to personnel while holding all other expenditures constant, because that would exceed the budget limit by one dollar.

Within the reallocation scenario, there are two variations. First, each manager has the same budget. Here, the predictors are linearly dependent, and the analysis must accommodate dependencies (Davison et al., 2023). In the second scenario, managers’ budgets vary. The referent and ordered approaches model resource reallocations. In the referent model, each regression weight for variable 2 − V is the expected increase in Y obtained by transferring a unit from Variable 1 to Variable v holding all other predictors (and total) constant. The first predictor (the sum) represents the contribution of the budget total for a particular manager and accounts for the variation in Y attributable to variation in budget limits. The ordered equation approach models transfer from Variable v − 1 to Variable v, with the sum in the first predictor accounting for the effect of individual differences in budget total on Y.

The example above involves the allocation of financial predictors, but the allocation need not be financial. The predictors may represent time. A student may spend time in three reading activities: reading group, seat work assignments, and independent reading. The total time may vary by student, but given a particular total the student divides time between the activities. Terms can be added to test hypotheses about interactions: for example, independent reading and total reading time. The researcher may be interested in the multicollinearity of the model as the several sums in the ordered model will be correlated. The researcher may also be interested in the effect on R² of dropping a predictor. The effect of dropping one predictor from the referent or ordered model would generally be different than the effect of dropping the same predictor from the ordinary model, because other predictors in these models differ from those in the ordinary model.

An allocation model is potentially useful for many psychological/educational concepts (e.g., attention span Bradbury, 2016; McClelland et al., 2013). Attention span is often conceived as a limit on time of attention. Attention can be allocated to different foci. On a math story problem, attention could be allocated to time to read the scenario, time allocated to the accompanying graph, time for the response alternatives, and so on. The predictors could be measured using eye tracking. Other concepts that might be modeled with an allocation framework include working memory (Cowan, 2005; Oberauer et al., 2016) and patience (Schnitker, 2012).

Methods: Simulation Studies

We compared Type I errors and power of the F- and t-statistics utilizing substitution to that of an alternative involving a penalty function, the LM approach.

LM Approach

Since the F-statistic of Equation 7 requires the estimation of the constrained and full model, it may be more convenient to use $χ^{2}$ alternative tests, such as the Wald (W) test, the likelihood ratio (LR) test, or the LM test. The current study chose the LM test for two reasons. The LM test is easier to compute than the others because it only requires the estimation of the full model (shown in the following), and the three test statistics satisfy the following inequality: $W \geq L R \geq L M$ . In other words, among all three tests, the LM test is the most conservative, so that whenever the LM test rejects, so will the others (Engle, 1984). LM starts at the null hypothesis (e.g., $H_{0} : b_{1} = b_{2} = b_{3} = b_{4}$ ) and asks whether movement toward the alternative is an improvement. Under H ₀, the test statistic is asymptotically approximated by χ² (Evans & Savin, 1982). Rewriting H ₀ as a set of m linear constraints on the coefficients in the matrix form, $R b = r$ , where R is $m \times v$ , r is $m \times 1$ , and b is a v x 1 vector of regression weights. For the ordered model,

R = [\begin{matrix} 1 & - 1 & 0 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & 0 & 1 & - 1 \end{matrix}], b = [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}] and r = 0.

The χ² statistic of the LM test is:

\frac{{(R \hat{b} - r)}^{'} {(R {(X^{'} X)}^{- 1} R^{'})}^{- 1} (R \hat{b} - r)}{σ^{2}} \sim χ^{2} (m) .

where $\hat{b} = {(X^{'} X)}^{- 1} X' y$ contains the regression coefficients of the unconstrainted model. LM can also be used to test pairs of adjacent predictors, for example, $H_{0} : b_{1} = b_{2}$ , with the linear constraint

R = [\begin{matrix} 1 & - 1 & 0 & 0 \end{matrix}], b = [\begin{matrix} b_{1} \\ b_{2} \\ b_{3} \\ b_{4} \end{matrix}] and r = 0.

When LM is applied to a linear regression with two predictors, the global hypothesis equals the pairwise comparison (i.e., $H_{0} : b_{1} = b_{2}$ ). The denominator of the LM statistic ( $σ^{2})$ is unknown and usually estimated with the sample residual variance. Since LM is asymptotically distributed as $χ^{2}$ , it may yield misleading results in small samples (for details, see Evans & Savin, 1982). For that reason, Kiviet (1986) recommended the F-test of Equation 7 for small samples.

The simulation consists of two studies, one to study Type I errors when all regression weights are equal. We studied Type 1 errors both for the global hypothesis $H_{0} :$ $b_{1} = b_{2} = b_{3} \dots = b_{V} = b$ and for contrasts between adjacent coefficients: for example, $H_{0} : b_{2} - b_{1}$ = 0 or $b_{3} - b_{2}$ = 0 or $b_{4} - b_{3}$ = 0 (V − 1 pairwise hypotheses for V predictors). We also studied power for both tests. Regression weights are a function of two other statistics, the correlation of predictors with the criteria, $r_{x y}$ and the intercorrelations of predictors $r_{x x'}$ . Rather than manipulating regression weights directly, we manipulated them indirectly by varying $r_{x y}$ and $r_{x x'}$ to assess the Type I error rate and power under various combinations of $r_{x y}$ and $r_{x x'}$ .

Type I Error Rates

We studied Type I errors for tests of the global and contrast hypotheses for analysis type (substitution vs. LM), $r_{x y}$ (0.0, 0.3, and 0.6), $r_{x x'}$ (0.0, 0.3, 0.6), number of predictors (2 and 4), and sample size (40, 100, 500, and 1,000). Conditioning on the number of predictors, the other manipulated factors produced two factorial designs with $2 \times 3 \times 3 \times 4 =$ 72 cells. In each cell, all regression weights are equal, but across combinations of $r_{x y}$ and $r_{x x'}$ , they vary. Analysis type was a factor with two levels, substitution and LM. Predictor/criterion correlation $r_{x y}$ was a factor with three levels: 0.0, 0.3, and 0.6. Within a cell, $r_{x y}$ was equal for all predictors. Likewise, predictor intercorrelation $r_{x x'}$ was a factor with three levels: 0.0, 0.3, and 0.6. Within a cell, $r_{x x'}$ was the same for all pairs of predictors. The population (true) regression weights for the various combinations of $r_{x y}$ and $r_{x x'}$ are shown in Table 1.

Table 1.

Population Regression Weight Common to the Two and Four Predictors with Each Combination of Predictor Correlation r _xx' and Predictor/Criterion Correlation r _xy in the Study of Type I Errors

Two Predictors
	$r_{x x'} = 0.0$	$r_{x x'} = 0.3$	$r_{x x'} = 0.6$
$r_{x y} = 0.0$	.00	.00	.00
$r_{x y} = 0.3$	.30	.23	.19
$r_{x y} = 0.6$	.60	.46	.38
Four Predictors
$r_{x y} = 0.0$	.00	.00	.00
$r_{x y} = 0.3$	.30	.16	.11
$r_{x y} = 0.6$	.60	.32	.21

Within each cell, there were 1,000 replications. For each replication, the predictors and criterion were sampled from a multivariate normal distribution MVN (0, Σ). In Σ, all predictor/criterion correlations were equal to a value $r_{x y}$ specified by the experimental condition, and all predictor intercorrelations were set equal to $r_{x x'}$ specified by the experimental condition. Variances of all variables were set to 1. Table 1 shows the regression weight in the original form of the linear model (Equation 2a) for each combination of $r_{x y}$ and $r_{x x'}$ . The weights tend to increase with $r_{x y}$ and stay the same or decrease as $r_{x x'}$ increases. The weights for four predictors tend to be the same or smaller than those for two. In the various conditions, the regression weight ranges from 0.0 to 0.6. The significance level for the hypothesis test was .05.

Power

The design for the power study was the same as for the Type I error study with two changes. First, $r_{x y}$ was replaced by a mean $r_{x y}$ condition ( ${\bar{r}}_{x y})$ . For $r_{x y} = 0.0$ , there were two or four predictors with mean $r_{x y} = 0.0$ . For $r_{x y} = 0.3$ , there were two or four predictors with mean $r_{x y} = 0.3.$ Second, in the power study, there was another factor, $r_{x y}$ difference ( $Δ r_{x y})$ with four levels: 0.05, 0.1, 0.2, and 0.3. $Δ r_{x y}$ refers to the size of the difference between $r_{x y}$ for adjacent predictors. For instance, if there were two predictors and a mean $r_{x y} = 0.3$ and $Δ r_{x y} = 0.1$ , the two values of $r_{x y}$ are 0.25 and 0.35. These two correlations have mean $r_{x y} = 0.3$ and $Δ r_{x y} = 0.1$ . If there were four predictors and a mean $r_{x y} = 0.3$ and $Δ r_{x y} = 0.1$ , the four values of $r_{x y}$ were 0.15, 0.25, 0.35, and 0.45. These four values have mean $r_{x y} = 0.3$ and adjacent values have $Δ r_{x y} = 0.1$ . The extra factor, $Δ r_{x y}$ was not completely crossed with the other conditions, because the completely crossed design would have impossible cells. For instance, when mean $r_{x y} = 0.6$ and $Δ r_{x y} = 0.3$ , the four values of $r_{x y}$ would be $r_{x y}$ = 0.15, 0.45, 0.75, and 1.05, but a correlation greater than 1.0 is impossible. Thus, there are three cells in the design that involve a contradiction, the three cells corresponding to ${\bar{r}}_{x y} = 0.6$ and $Δ r_{x y} = 0.3$ ; see the last row of Table 3.

The two dependent variables for this study were power for the global hypothesis $H_{0} :$ $b_{1} = b_{2} = b_{3} \dots = b_{V} = b$ and the pairwise regression weight contrasts $H_{0} : b_{v} - b_{v - 1} = 0$ . The “true” regression weights are shown in Tables 2 and 3 for two and four predictors, respectively.

For instance, when there were four predictors, mean $r_{x y} = .3$ , $Δ r_{x y}$ = .1, and the off-diagonal element $r_{x x^{'}} = .6$ , the true regression weights for the four variables were −0.27, −0.02, 0.23, and 0.48. In this same condition for two variables, the true regression weights were 0.06 and 0.31.

Table 2.

Power Study Two Predictors: Population Regression Weights for Each Combination of Predictor Correlation $r_{xx}$ and Mean Predictor/Criterion Correlation $r_{xy}$ and $r_{xy}$ Difference in Study of Power (Two Predictors)

Mean $r_{x y}$	$Δ r_{x y}$	$r_{x x'} = 0$	$r_{x x'} = 0.3$	$r_{x x'} = 0.6$
.0	.05	(−.03, .03)	(−.04, .04)	(−.06, .06)
.0	.1	(−.05, .05)	(−.07, .07)	(−.13, .13)
.0	.2	(−.10, .10)	(−.14, .14)	(−.25, .25)
.0	.3	(−.15, .15)	(−.21, .21)	(−.38, .38)
.3	.05	(.28, .33)	(.20, .27)	(.13, .25)
.3	.1	(.25, .35)	(.16, .30)	(.06, .31)
.3	.2	(.2, .4)	(.09, .37)	(.06, .44)
.3	.3	(.15, .45)	(.02, .45)	(−.19, .56)
.6	.05	(.58, .63)	(.43, .50)	(.31, .44)
.6	.1	(.55, .65)	(.39, .58)	(.25, .5)
.6	.2	(.5, .7)	(.32, .60)	(.13, .63)
.6	.3	(.45, .75)	(.25, .68)	(0, .75)

For both the Type I and Power portions of the study, we fitted the ordered substitution model and used the tests of null hypotheses for the regression weights in that model to test whether regression weight contrasts in the original model equaled 0. Type I and power from fitting the substitution model were then compared to those for the LM.

Results: Type I Errors

Type I error rates for each condition are in Tables 4 and 5. ANOVA results on the experimental factors for the simulation study are reported selectively for larger effects. For two predictors (Table 4), Type I errors were acceptable for both the global and pairwise hypotheses under all simulation conditions (Bradley, 1978). In the ANOVA, the only statistically significant effect (p < .05) was sample size, and it accounted for little variance as measured by the effect size measure eta squared, $η^{2} = .001$ . The Type I error rates for the F-statistic are consistently smaller (for smaller samples) or equal to (larger samples) those for the LM statistic, but the differences are small and of little practical importance.

Table 3.

Power Study Four Predictors: Population Regression Weights for Each Combination of Predictor Correlation $r_{xx}$ and Mean Predictor/Criterion Correlation $r_{xy}$ and $r_{xy}$ Difference in Study of Power

Mean $r_{x y}$	$Δ r_{x y}$	$r_{x x'} = 0.0$	$r_{x x'} = 0.3$	$r_{x x'} = 0.6$
.0	.05	(−0.08, −0.03, 0.03, 0.08)	(−0.11, −0.04, 0.04, 0.11)	(−0.19, −0.06, 0.06, 0.19)
.0	.1	(−0.15, −0.05, 0.05, 0.15)	(−0.21, −0.07, 0.07, 0.21)	(−0.38, −0.13, 0.13, 0.38)
.0	.2	(−0.3, −0.1, 0.1, 0.3)	(−0.43, −0.14, 0.14, 0.43)	(−0.75, −0.25, 0.25, 0.75)
.0	.3	(−0.45, −0.15, 0.15, 0.45)	(−0.64, −0.21, 0.21, 0.64)	(−1.13, −0.38, 0.38, 1.13)
.3	.05	(0.23, 0.28, 0.33, 0.38)	(0.05, 0.12, 0.19, 0.27)	(−0.08, 0.04, 0.17, 0.29)
.3	.1	(0.15, 0.25, 0.35, 0.45)	(−0.06, 0.09, 0.23, 0.37)	(−0.27, −0.02, 0.23, 0.48)
.3	.2	(0, 0.2, 0.4, 0.6)	(−0.27, 0.02, 0.3, 0.59)	(−0.64, −0.14, 0.36, 0.86)
.3	.3	(−0.15, 0.15, 0.45, 0.75)	(−0.48, −0.06, 0.37, 0.80)	(−.102, −0.27, 0.48, 1.23)
.6	.05	(0.53, 0.58, 0.63, 0.68)	(0.21, 0.28, 0.35, 0.42)	(0.03, 0.15, 0.28, 0.40)
.6	.1	(0.45, 0.55, 0.65, 0.75)	(0.1, 0.24, 0.39, 0.53)	(−0.16, 0.09, 0.34, 0.59)
.6	.2	(0.3, 0.5, 0.7, 0.9)	(−0.11, 0.17, 0.46, 0.74)	(−0.54, −0.04, 0.46, 0.96)
.6	.3	NA	NA	NA

Similar results were observed for four predictors (Table 5). Type I errors were acceptable for all conditions (Bradley, 1978). The only significant effect in the ANOVA was sample size, but that was small ( $η^{2} < .001)$ . Again, the LM approach was slightly more likely to reject than substitution. The differences became smaller (or vanished) as sample size increased. A possible explanation is that the LM statistic asymptotically approaches the $χ^{2}$ distribution, leading to rejection rates for the LM statistic closer to the nominal rate of .05 as sample size increases.

Results: Power

Power for two predictors is in Tables 6 (LM) and 7 (substitution). In the ANOVA, when the regression had two or four predictors, the largest effects were for sample size and $Δ r_{x y}$ , both with $η^{2} > .09$ . Power increased with both sample size and $Δ r_{x y}$ . LM had slightly larger power than substitution with differences decreasing as sample size increased. Again, this may be due to the relationship of LM to $χ^{2}$ (Evans & Savin, 1982). With two predictors, the F-statistic for the global effect and the t-statistic for the pairwise effect are identical as t is the square-root of F. LM for the global and piecewise effects yielded virtually identical power. The test was underpowered (<.8) with sample size 40 for all but one condition: ${\bar{r}}_{x y} = .6,$ $r_{x x'} = .6,$ and $Δ r_{x y} = .3$ . Only in the condition with the largest ${\bar{r}}_{x y}$ , $r_{x x'}$ , and $Δ r_{x y}$ was power adequate with small sample size. The situation was little better with sample size 100. There were only six conditions with adequate power: ${\bar{r}}_{x y} = 0,$ $r_{x x'} = .6,$ and $Δ r_{x y} = .3; {\bar{r}}_{x y} = .3,$ $r_{x x'} =$ $.6,$ and $Δ r_{x y} = .3; {\bar{r}}_{x y} = 0.6,$ $r_{x x'} = 0,$ and $Δ r_{x y} = .3; {\bar{r}}_{x y} = .6,$ $r_{x x'} = .3,$ and $Δ r_{x y} = .3; {\bar{r}}_{x y} = .6,$ $r_{x x'} = .6,$ and $Δ r_{x y} = .2; {\bar{r}}_{x y} = .6,$ and $r_{x x'} = .6,$ and $Δ r_{x y} = .3.$ All but one of the six involved the largest effect $Δ r_{x y} = .3.$ Again, these figures reflect the fact that power tends to increase with increasing values of ${\bar{r}}_{x y}$ , $r_{x x'}$ , and $Δ r_{x y}$ .

Table 4.

Type I Error Rates for Two predictors With Each Combination of Simulation Conditions ^a

			Lagrange Multiplier	Substitution Approach
N	$r_{x x'}$	$r_{x y}$	$χ^{2}$	F
40	.0	.0	.062	.056
40	.0	.3	.062	.055
40	.0	.6	.068	.059
40	.3	.0	.062	.056
40	.3	.3	.059	.053
40	.3	.6	.068	.060
40	.6	.0	.062	.056
40	.6	.3	.059	.054
40	.6	.6	.067	.055
100	.0	.0	.052	.049
100	.0	.3	.051	.048
100	.0	.6	.048	.046
100	.3	.0	.052	.049
100	.3	.3	.052	.049
100	.3	.6	.050	.048
100	.6	.0	.052	.049
100	.6	.3	.051	.048
100	.6	.6	.049	.049
500	.0	.0	.066	.065
500	.0	.3	.064	.063
500	.0	.6	.060	.060
500	.3	.0	.066	.065
500	.3	.3	.064	.063
500	.3	.6	.061	.058
500	.6	.0	.066	.065
500	.6	.3	.064	.064
500	.6	.6	.060	.060
1,000	.0	.0	.049	.049
1,000	.0	.3	.049	.049
1,000	.0	.6	.048	.048
1,000	.3	.0	.049	.049
1,000	.3	.3	.050	.050
1,000	.3	.6	.050	.049
1,000	.6	.0	.049	.049
1,000	.6	.3	.051	.051
1,000	.6	.6	.052	.052

^a The global hypothesis is identical to the pairwise comparison (i.e., $H_{0} : b_{1} = b_{2}$ ), so the Type I error rate is the same and therefore not reported separately.

Table 5.

Type I Error Rates for Four Predictors With Each Combination of Simulation Conditions

			Lagrange Multiplier				Substitution Approach
N	$r_{x x'}$	$r_{x y}$	$b_{1} = b_{2}$	$b_{2} = b_{3}$	$b_{3} = b_{4}$	$χ^{2}$	$b_{1} = b_{2}$	$b_{2} = b_{3}$	$b_{3} = b_{4}$	F
40	0	0	.058	.046	.045	.062	.054	.041	.042	.035
40	0	0.3	.056	.057	.045	.060	.050	.047	.040	.041
40	0	0.6	.060	.046	.045	.058	.053	.044	.042	.045
40	0.3	0	.058	.046	.045	.062	.054	.041	.042	.035
40	0.3	0.3	.059	.054	.045	.058	.053	.048	.042	.042
40	0.3	0.6	.054	.054	.048	.060	.051	.049	.041	.046
40	0.6	0	.058	.046	.045	.062	.054	.041	.042	.035
40	0.6	0.3	.054	.053	.048	.059	.049	.043	.042	.037
40	0.6	0.6	.058	.056	.042	.061	.052	.048	.040	.041
100	0	0	.058	.044	.051	.052	.056	.044	.048	.050
100	0	0.3	.059	.048	.044	.057	.058	.045	.042	.051
100	0	0.6	.062	.057	.053	.054	.060	.054	.052	.051
100	0.3	0	.058	.044	.051	.052	.056	.044	.048	.050
100	0.3	0.3	.056	.044	.052	.053	.053	.043	.044	.050
100	0.3	0.6	.062	.051	.048	.057	.059	.050	.046	.052
100	0.6	0	.058	.044	.051	.052	.056	.044	.048	.050
100	0.6	0.3	.057	.045	.052	.053	.056	.043	.050	.050
100	0.6	0.6	.059	.049	.047	.057	.057	.046	.042	.053
500	0	0	.055	.056	.050	.053	.054	.054	.048	.053
500	0	0.3	.057	.055	.038	.060	.057	.055	.038	.059
500	0	0.6	.058	.050	.053	.051	.057	.047	.053	.050
500	0.3	0	.055	.056	.050	.053	.054	.054	.048	.053
500	0.3	0.3	.060	.055	.045	.058	.060	.055	.044	.058
500	0.3	0.6	.060	.050	.036	.061	.060	.050	.036	.061
500	0.6	0	.055	.056	.050	.053	.054	.054	.048	.053
500	0.6	0.3	.057	.056	.049	.056	.057	.056	.048	.056
500	0.6	0.6	.059	.055	.040	.060	.059	.054	.039	.059
1,000	0	0	.053	.060	.054	.053	.053	.060	.054	.052
1,000	0	0.3	.052	.065	.048	.053	.052	.064	.047	.053
1000	0	0.6	.060	.051	.046	.057	.060	.051	.046	.057
1,000	0.3	0	.053	.060	.054	.053	.053	.060	.054	.052
1,000	0.3	0.3	.054	.060	.053	.056	.054	.060	.053	.056
1,000	0.3	0.6	.051	.060	.048	.057	.050	.060	.048	.056
1,000	0.6	0	.053	.060	.054	.053	.053	.060	.054	.052
1,000	0.6	0.3	.054	.060	.053	.059	.054	.060	.053	.059
1,000	0.6	0.6	.053	.064	.050	.054	.053	.064	.049	.054

Table 6.

Statistical Power for Two predictors With the Lagrange Multiplier ( $χ^{2})$ ^a

			Sample Size (N)
$r_{x x'}$	$r_{x y}$	$Δ r_{x y}$	$40$	100	500	1,000
0	0	.05	.074	0.076	0.138	0.184
0	0	.1	.091	0.110	0.356	0.615
0	0	.2	.139	0.298	0.915	0.996
0	0	.3	.272	0.553	1.000	1.000
0	.3	.05	.062	0.085	0.137	0.238
0	.3	.1	.090	0.137	0.398	0.685
0	.3	.2	.177	0.343	0.935	1.000
0	.3	.3	.339	0.656	1.000	1.000
0	.6	.05	.068	0.102	0.318	0.535
0	.6	.1	.127	0.264	0.856	0.986
0	.6	.2	.405	0.761	1.000	1.000
0	.6	.3	.744	0.992	1.000	1.000
.3	0	.05	.072	0.082	0.161	0.273
.3	0	.1	.086	0.148	0.484	0.768
.3	0	.2	.181	0.393	0.961	1.000
.3	0	.3	.372	0.759	1.000	1.000
.3	.3	.05	.080	0.076	0.159	0.319
.3	.3	.1	.088	0.160	0.521	0.809
.3	.3	.2	.212	0.454	0.988	1.000
.3	.3	.3	.407	0.791	1.000	1.000
.3	.6	.05	.060	0.101	0.287	0.545
.3	.6	.1	.130	0.241	0.812	0.980
.3	.6	.2	.408	0.736	1.000	1.000
.3	.6	.3	.711	0.977	1.000	1.000
.6	0	.05	.075	0.088	0.230	0.393
.6	0	.1	.123	0.188	0.702	0.937
.6	0	.2	.280	0.625	0.998	1.000
.6	0	.3	.596	0.931	1.000	1.000
.6	.3	.05	.067	0.092	0.252	0.469
.6	.3	.1	.130	0.224	0.769	0.966
.6	.3	.2	.340	0.663	1.000	1.000
.6	.3	.3	.605	0.967	1.000	1.000
.6	.6	.05	.077	0.113	0.382	0.684
.6	.6	.1	.161	0.296	0.923	0.998
.6	.6	.2	.470	0.858	1.000	1.000
.6	.6	.3	.868	1.000	1.000	1.000

^a The global hypothesis is identical to the pairwise comparison (i.e., $H_{0} : b_{1} = b_{2}$ )

Table 7.

Statistical Power for Two predictors With the Substitution approach (F)^a

			Sample Size (N)
$r_{x x'}$	$r_{x y}$	$Δ r_{x y}$	$40$	100	500	1,000
0	0	.05	.063	0.071	0.135	0.184
0	0	.1	.078	0.107	0.355	0.615
0	0	.2	.124	0.286	0.914	0.996
0	0	.3	.253	0.541	1.000	1.000
0	.3	.05	.056	0.081	0.137	0.237
0	.3	.1	.078	0.125	0.398	0.683
0	.3	.2	.154	0.335	0.934	1.000
0	.3	.3	.321	0.649	1.000	1.000
0	.6	.05	.063	0.096	0.313	0.534
0	.6	.1	.115	0.256	0.853	0.985
0	.6	.2	.379	0.753	1.000	1.000
0	.6	.3	.730	0.992	1.000	1.000
.3	0	.05	.064	0.078	0.161	0.272
.3	0	.1	.074	0.141	0.484	0.764
.3	0	.2	.164	0.389	0.960	1.000
.3	0	.3	.341	0.752	1.000	1.000
.3	.3	.05	.072	0.070	0.158	0.318
.3	.3	.1	.078	0.155	0.518	0.809
.3	.3	.2	.192	0.438	0.988	1.000
.3	.3	.3	.382	0.787	1.000	1.000
.3	.6	.05	.054	0.093	0.286	0.544
.3	.6	.1	.122	0.234	0.810	0.980
.3	.6	.2	.380	0.723	1.000	1.000
.3	.6	.3	.688	0.973	1.000	1.000
.6	0	.05	.066	0.087	0.229	0.392
.6	0	.1	.110	0.183	0.699	0.937
.6	0	.2	.259	0.619	0.998	1.000
.6	0	.3	.572	0.924	1.000	1.000
.6	.3	.05	.055	0.088	0.249	0.469
.6	.3	.1	.118	0.220	0.768	0.966
.6	.3	.2	.312	0.653	1.000	1.000
.6	.3	.3	.587	0.966	1.000	1.000
.6	.6	.05	.067	0.104	0.381	0.683
.6	.6	.1	.141	0.293	0.922	0.998
.6	.6	.2	.449	0.852	1.000	1.000
.6	.6	.3	.859	1.000	1.000	1.000

^aThe global hypothesis is identical to the pairwise comparison (i.e., $H_{0} : b_{1} = b_{2}$ )

With sample size 500, the global and pairwise tests were adequately powered for all conditions, where $Δ r_{x y} = .2$ or .3, and some conditions, where $Δ r_{x y} = .1$ . and ${\bar{r}}_{x y} = .6$ . The picture was much the same for samples of 1,000: adequate power for all conditions in which $Δ r_{x y} = .2$ or .3 and some conditions where $Δ r_{x y} = .1$ and ${\bar{r}}_{x y} = .6$ . For none of the scenarios and sample sizes studied was the global test able to adequately detect the smallest $Δ r_{x y}$ = .05. Power for four predictors is shown in Tables 8 (LM) and 9 (substitution). $Δ r_{x y}$ was fixed between adjacent pairs of regression weights for each condition, and the power for the pairwise test should be similar for each adjacent pair. To save space, we calculated average power of the three pairwise hypotheses and list it under the column “Avg. Pairwise.” For four predictors and sample size 40, the global test was adequately powered for $Δ r_{x y} = .2$ or .3, with a few exceptions involving ${\bar{r}}_{x y}$ = 0 and/or $r_{x x'} = 0$ . With sample size 100, the global test had adequate powered for $Δ r_{x y}$ = .1. With sample size 500, it was adequately powered for all conditions except a few involving the smallest $Δ r_{x y}$ = .05 plus ${\bar{r}}_{x y}$ = 0 and/or $r_{x x'} = 0$ . With sample sizes of 1,000, the global test is adequately powered for all conditions in this study. As compared to the global test with two predictors, that with four predictors has more power.

Table 8.

Statistical Power for Four predictors With the Lagrange Multiplier ( $χ^{2}$ )^a

			Sample Size (N)
$r_{x x'}$	$r_{x y}$	$Δ r_{x y}$	$40$		100		500		1,000
$r_{x x'}$	$r_{x y}$	$Δ r_{x y}$	Avg. Pairwise	Global	Avg. Pairwise	Global	Avg. Pairwise	Global	Avg. Pairwise	Global
0	0	.05	0.059	0.085	0.068	0.137	0.115	0.530	0.202	0.852
0	0	.1	0.071	0.215	0.118	0.464	0.371	0.997	0.626	1.000
0	0	.2	0.163	0.691	0.348	0.989	0.940	1.000	0.999	1.000
0	0	.3	0.402	0.995	0.787	1.000	1.000	1.000	1.000	1.000
0	.3	.05	0.071	0.112	0.065	0.166	0.174	0.759	0.290	0.974
0	.3	.1	0.086	0.269	0.149	0.665	0.526	1.000	0.820	1.000
0	.3	.2	0.254	0.920	0.547	0.999	0.996	1.000	1.000	1.000
0	.3	.3	0.817	1.000	0.997	1.000	1.000	1.000	1.000	1.000
0	.6	.05	0.067	0.120	0.066	0.135	0.153	0.647	0.243	0.923
0	.6	.1	0.080	0.201	0.110	0.459	0.385	0.998	0.639	1.000
0	.6	.2	0.127	0.503	0.232	0.903	0.766	1.000	0.972	1.000
0	.6	.3	NA	NA	NA	NA	NA	NA	NA	NA
.3	0	.05	0.066	0.104	0.072	0.188	0.145	0.693	0.270	0.958
.3	0	.1	0.088	0.272	0.147	0.619	0.496	1.000	0.796	1.000
.3	0	.2	0.224	0.879	0.498	1.000	0.995	1.000	1.000	1.000
.3	0	.3	0.710	1.000	0.981	1.000	1.000	1.000	1.000	1.000
.3	.3	.05	0.064	0.102	0.086	0.219	0.191	0.827	0.323	0.991
.3	.3	.1	0.103	0.336	0.164	0.687	0.588	1.000	0.865	1.000
.3	.3	.2	0.297	0.941	0.633	1.000	0.999	1.000	1.000	1.000
.3	.3	.3	0.942	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.3	.6	.05	0.087	0.278	0.142	0.610	0.513	1.000	0.799	1.000
.3	.6	.1	0.249	0.891	0.519	1.000	0.994	1.000	1.000	1.000
.3	.6	.2	0.987	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.3	.6	.3	NA	NA	NA	NA	NA	NA	NA	NA
.6	0	.05	0.067	0.124	0.083	0.281	0.238	0.927	0.443	1.000
.6	0	.1	0.112	0.443	0.223	0.867	0.756	1.000	0.967	1.000
.6	0	.2	0.476	0.997	0.862	1.000	1.000	1.000	1.000	1.000
.6	0	.3	0.997	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.6	.3	.05	0.068	0.156	0.086	0.317	0.284	0.965	0.487	1.000
.6	.3	.1	0.134	0.527	0.247	0.922	0.819	1.000	0.981	1.000
.6	.3	.2	0.578	1.000	0.945	1.000	1.000	1.000	1.000	1.000
.6	.3	.3	0.875	1.000	0.999	1.000	1.000	1.000	1.000	1.000
.6	.6	.05	0.081	0.236	0.133	0.571	0.452	1.000	0.743	1.000
.6	.6	.1	0.214	0.832	0.440	1.000	0.987	1.000	1.000	1.000
.6	.6	.2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.6	.6	.3	NA	NA	NA	NA	NA	NA	NA	NA

^a Avg. Pairwise stands for the average power of three pairwise null hypotheses ( $b_{1} = b_{2}, b_{2} = b_{3}$ , and $b_{3} = b_{4})$ , and Global stands for the power of the global null hypothesis ( $b_{1} = b_{2} = b_{3} = b_{4}) .$

Table 9.

Statistical Power for Four Predictors With the Substitution Approach (F)^a

			Sample Size (N)
$r_{x x'}$	$r_{x y}$	$Δ r_{x y}$	$40$		100		500		1,000
$r_{x x'}$	$r_{x y}$	$Δ r_{x y}$	Avg. Pairwise	Global	Avg. Pairwise	Global	Avg. Pairwise	Global	Avg. Pairwise	Global
0	0	.05	0.052	0.099	0.067	0.139	0.115	0.530	0.202	0.853
0	0	.1	0.065	0.225	0.114	0.470	0.369	0.997	0.626	1.000
0	0	.2	0.151	0.706	0.342	0.989	0.939	1.000	0.999	1.000
0	0	.3	0.385	0.997	0.780	1.000	1.000	1.000	1.000	1.000
0	.3	.05	0.064	0.122	0.062	0.170	0.173	0.761	0.289	0.974
0	.3	.1	0.079	0.287	0.142	0.668	0.524	1.000	0.819	1.000
0	.3	.2	0.237	0.927	0.536	0.999	0.996	1.000	1.000	1.000
0	.3	.3	0.801	1.000	0.996	1.000	1.000	1.000	1.000	1.000
0	.6	.05	0.061	0.126	0.062	0.140	0.152	0.646	0.243	0.923
0	.6	.1	0.069	0.222	0.105	0.473	0.383	0.998	0.639	1.000
0	.6	.2	0.113	0.530	0.225	0.904	0.764	1.000	0.972	1.000
0	.6	.3	NA	NA	NA	NA	NA	NA	NA	NA
.3	0	.05	0.057	0.115	0.068	0.196	0.145	0.693	0.269	0.959
.3	0	.1	0.077	0.285	0.144	0.626	0.495	1.000	0.795	1.000
.3	0	.2	0.208	0.884	0.489	1.000	0.995	1.000	1.000	1.000
.3	0	.3	0.693	1.000	0.979	1.000	1.000	1.000	1.000	1.000
.3	.3	.05	0.056	0.106	0.084	0.226	0.189	0.828	0.323	0.991
.3	.3	.1	0.092	0.356	0.159	0.698	0.587	1.000	0.865	1.000
.3	.3	.2	0.279	0.944	0.626	1.000	0.999	1.000	1.000	1.000
.3	.3	.3	0.937	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.3	.6	.05	0.080	0.301	0.139	0.616	0.511	1.000	0.797	1.000
.3	.6	.1	0.230	0.905	0.510	1.000	0.994	1.000	1.000	1.000
.3	.6	.2	0.986	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.3	.6	.3	NA	NA	NA	NA	NA	NA	NA	NA
.6	0	.05	0.060	0.135	0.079	0.288	0.237	0.927	0.443	1.000
.6	0	.1	0.103	0.474	0.217	0.876	0.755	1.000	0.967	1.000
.6	0	.2	0.455	0.997	0.857	1.000	1.000	1.000	1.000	1.000
.6	0	.3	0.996	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.6	.3	.05	0.061	0.169	0.082	0.320	0.282	0.965	0.487	1.000
.6	.3	.1	0.122	0.549	0.241	0.925	0.818	1.000	0.981	1.000
.6	.3	.2	0.549	1.000	0.942	1.000	1.000	1.000	1.000	1.000
.6	.3	.3	0.863	1.000	0.999	1.000	1.000	1.000	1.000	1.000
.6	.6	.05	0.073	0.255	0.128	0.578	0.451	1.000	0.742	1.000
.6	.6	.1	0.200	0.846	0.428	1.000	0.987	1.000	1.000	1.000
.6	.6	.2	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
.6	.6	.3	NA	NA	NA	NA	NA	NA	NA	NA

^aAvg. Pairwise stands for the average power of three pairwise null hypotheses ( $b_{1} = b_{2}, b_{2} = b_{3}$ , and $b_{3} = b_{4})$ , and Global stands for the power of the global null hypothesis ( $b_{1} = b_{2} = b_{3} = b_{4}) .$

The pairwise statistics are less well powered than the global tests, given equal sample size. With a sample size of 40, the pairwise test was generally adequately powered for large effects of $Δ r_{x y}$ = .3, except when ${\bar{r}}_{x y}$ = 0 and/or $r_{x x'}$ = 0. It was adequately powered to detect $Δ r_{x y}$ = .2 if ${\bar{r}}_{x y}$ = .6 and $r_{x x'}$ = .6. With a sample size of 100, it was adequately powered to detect only the largest effect $Δ r_{x y}$ = .3 with a few exceptions where it was adequately powered to detect $Δ r_{x y}$ = .2 with ${\bar{r}}_{x y}$ = .6. With sample sizes of 500, it was adequately powered to detect effects of $Δ r_{x y}$ = .3 or $Δ r_{x y}$ = .2, and sometimes adequately powered to detect $Δ r_{x y}$ = .1 if ${\bar{r}}_{x y}$ = .6. Sample size 1,000 was adequately powered to detect effects of $Δ r_{x y}$ = .3 or $Δ r_{x y}$ = .2 in all conditions and was adequately powered to detect the effects of $Δ r_{x y}$ = .1 if ${\bar{r}}_{x y} \geq .3$ . Power is a complex function of sample size and $Δ r_{x y}$ ( $η^{2}$ s were larger than .15) with mean $r_{x y}$ and $r_{x x'}$ playing lesser roles ( $η^{2}$ s were less than .03).

With four predictors, global power was adequate for many conditions. The modest power of the pairwise tests did not always permit identification of the pairs that differed. The LM displayed slightly higher power for small samples. The differences disappear in larger samples. In smaller samples, the LM has slightly higher Type I errors but also slightly higher power. Thus, the LM approach was more prone to reject whether or not the null hypothesis was true for small samples.

High School Coursework and Achievement Example

To illustrate the ordered method, we used data from the High School Longitudinal Study (HSLS-09; Ingels et al., 2015; Ingels et al., 2014). There are 10 predictors corresponding to 10 categories of high school math courses shown in Table 10 (e.g., Algebra 1, Geometry, Algebra 2, etc.). The criterion is high school math achievement score. Our data are a subset of a larger dataset. Our sample size was 2,900, after incorporating a design effect of 4.0 to account for the cluster nature of the sample. Cluster samples take subjects within clusters who tend to be more homogenous, and thus, the number of subjects overestimates the independent information in the sample (Kish & Frankel, 1974). The sample size was reduced by a factor of 4.0 to compensate for dependencies arising from the cluster sampling. Each predictor is measured in the same unit; the number of Carnegie Units taken (CUs) by the student in a coursework area (Ingels et al., 2015). A score of 1 in high school algebra means the student completed 1 CU of Algebra 1. Based on course content and other factors, the National Center for Educational Statistics (NCES) rank ordered the courses, and the ordering is shown from low to high in Column 2 of Table 10. Lee et al. (1998) found that high school math achievement test scores were related to the level of the highest math course reached by the student. For our analysis, we were interested in whether a CU is associated with the same expected increase in math achievement regardless of the course in which it was earned. Specifically, we were interested in whether higher level courses were associated with larger expected achievement scores. Thus, we were interested in whether qualitative differences in course content moderate the effect of one CU on math achievement.

We fit four models to the data. As a baseline, we fit the reduced model of Equation 6 with a single predictor $T_{1} = \sum_{v = 1}^{v = 10} X_{v}$ . T ₁ is the total number of math CUs taken by the student in high school. This corresponds to the submodel in which all 10 predictors have the same regression weight, and one CU of math is associated with the same expected increase in math achievement irrespective of the course. For this model, $R^{2}$ and the adjusted R ²=.215. Next, we fit the full linear regression model of Equation 2a with 10 predictors

A c h i e v e m e n t s c o r e = b_{1} X_{1} + b_{2} X_{2} + \dots + b_{10} X_{10} + a + e .

Results of this second analysis are shown in Table 10. The first column shows the ten course categories ordered by NCES’s level of coursework. The third column shows the empirical estimate of the regression weight and statistical significance. All regression weights are significantly different from zero (p < .05). In what may seem counterintuitive, the regression weight for Algebra 1 is negative. Algebra 1 indicates whether the student took Algebra 1 in high school. Having delayed taking Algebra 1 until high school detracts from a student’s expected level of achievement, hence the negative regression weight for Algebra 1. For this full model, R ²=.396 and the adjusted R ²=.394. As compared to the submodel in which all weights are constrained equal, the full model almost doubles the variance accounted for (.215 vs. .396).

Table 10.

Regression Weights for Models Predicting Math Achievement From Coursework

	Full Model		Referent Model		Ordered Model
	Weight	Value	Weight	Value	Weight	Value
Intercept		53.081***		53.081***		53.081***
Algebra 1	b₁	−1.137*
Geometry	b₂	1.705**	$b_{2} - b_{1}$	2.843**
Algebra 2	b₃	5.453***	$b_{3} - b_{1}$	6.59***
Trigonometry	b₄	7.602***	$b_{4} - b_{1}$	8.739***
Adv. Math	b₅	2.235***	$b_{5} - b_{1}$	3.372***
Statistics	b₆	6.052***	$b_{6} - b_{1}$	7.189***
AP IB Math	b₇	8.809***	$b_{7} - b_{1}$	9.946***
Precalculus	b₈	13.139***	$b_{8} - b_{1}$	14.277***
Calculus	b₉	10.797***	$b_{9} - b_{1}$	11.934***
AP IB Calculus	b₁₀	14.544***	$b_{10} - b_{1}$	15.681***	$b_{10} - b_{9}$	3.747**
T₁			b ₁	−1.137*	b ₁	−1.137*
T₂					$b_{2} - b_{1}$	2.843***
T₃					$b_{3} - b_{2}$	3.748***
T₄					$b_{4} - b_{3}$	2.149**
T₅					$b_{5} - b_{4}$	−5.367***
T₆					$b_{6} - b_{5}$	3.817**
T₇					$b_{7} - b_{6}$	2.757
T₈					$b_{8} - b_{7}$	4.331**
T₉					$b_{9} - b_{8}$	−2.342
R²		.396		.396		.396
Adjusted R²		.394		.394		.394

Note. Adv. Math = other advanced math; AP IB = advanced placement/international baccalaureate course eligible for college credit.

Source. U.S. Department of Education, National Center for Education Statistics, High School Longitudinal Study of 2009 (HSLS:09) Base Year (2009), First Follow-up (2012, 2013), Transcript (2013, 2014).

*p < .05. **p < .01. ***p < .001.

Next, we tested the global hypothesis: $H_{0} : b_{1} = b_{2} = b_{3} = b_{4} = b_{5} = b_{6} = b_{7} = b_{8} = b_{9} = b_{10}$ by computing the F-statistic in Equation 7, where the full model is that of Equation 11 and the reduced model (Equation 6) has a single predictor T ₁. $F = 96.205, p < .001$ , which leads to the rejection of the hypothesis that all 10 regression weights are equal. Qualitative differences in course content moderate the impact of a CU on expected math achievement. This fact may not be due to the coursework itself but to differences in the students who take the different courses.

Third, we fit the referent model designating Algebra 1 as the reference variable and all others as focal variables:

A c h i e v e m e n t s c o r e = b_{1} T_{1} + (b_{2} - b_{1}) X_{2} + (b_{3} - b_{1}) X_{3} + \dots + (b_{10} - b_{1}) X_{10} + a + e .

This model provides a test that one CU of each focal course yields the same increase in expected math achievement as does one unit of Algebra 1. This model was fit by regressing the achievement test scores onto $T_{1}, X_{2}, X_{3}, X_{4}, X_{5}, X_{6}, X_{7}, X_{8}, X_{9}, and X_{10}$ . Here, we substitute T ₁ for X ₁in the original model. Columns 4 and 5 of Table 10 refer to this analysis. Column 4 shows the regression weights for the focal variables in this model as a contrast of the weights in Equation 11, and Column 5 shows the value of that contrast. As expected, the weight on the first predictor X ₁ for Equation 11 equals that for the first predictor T ₁ in the referent reparameterization (Equation 12), −1.137. Each weight in the reparameterization, except the first, is a difference between two weights in Equation 11. In the reparameterization, the weight on X ₂ (Geometry) is 2.843, the difference between the weights for Geometry and Algebra 1 (the reference variable) in Equation 11: 1.705 – (−1.137) within rounding. Likewise, the weight for $X_{3}$ (Equation 12) is 6.590, and the difference of the weights for Algebra 2 and Algebra 1 in Equation 11: 5.453 – (−1.137). The referent model has the same R ²=.396 and adjusted R ²=.394 as the original, because it is simply a reparameterization. In Table 10, the t test of regression weights led to rejection of the null hypothesis $H_{0} : b_{v} = b_{1}$ for all focal variables ( $p < .05)$ . All of the other courses have significantly higher regression weights than does Algebra 1.

Finally, we fitted the ordered reparameterization, the model in Equation 4c:

A c h i e v e m e n t s c o r e = b_{1} T_{1} + (b_{2} - b_{1}) T_{2} + (b_{3} - b_{2}) T_{3} + \dots + (b_{10} - b_{9}) X_{10} + a + e .

To fit Equation 13, we regressed the achievement test score onto $T_{1}, T_{2}, T_{3}, T_{4}, T_{5}, T_{6}, T_{7}, T_{8}, T_{9}, and X_{10} .$ Columns 6 and 7 of Table 10 refer to this analysis. Column 6 shows the weights in this model as a function of the weights in the original model; each being a pairwise contrast. Column 7 shows the value of that contrast. Again, the weight on X ₁ in Equation 11 equals that of T ₁ in Equation 13, −1.137. The other weights in Equation 13 are contrasts between two consecutive weights from Equation 11. In the reparameterization, the weight on T ₂ (Geometry) is 2.843, the difference between the weights for Geometry X ₂ and Algebra $1 X_{1}$ in the original model: 1.705 – (−1.137) = 2.842 (within rounding). Likewise, the weight for T ₃ is 3.748, the difference between the weights for Geometry $X_{3}$ and Algebra 2 X ₂ in the original model: 5.453 − 1.705 = 3.748. The ordered model has the same R ²=.396 and adjusted R ²=.394 as the original and referent models, since it is simply a reparameterization. The t-tests support the given order for the courses for all except 3 consecutive pairs. T ₅ (p < .05) is significant, which corresponds to the difference for other advanced math and trigonometry, but the weight is negative suggesting trigonometry (lower in the ordering), has a significantly higher regression coefficient than other advanced math. Other advanced math may be a catch-all category that may be less rigorous than first thought. In the ordered model, the coefficients for $T_{7}$ and T ₉ were not significant, suggesting no significant difference between coefficients for advanced placement/international baccalaureate math and statistics nor between calculus and precalculus.

The substitution models in Equations 12 and 13 may or may not mirror student’s decision process, and the model need not do so to be of interest. However, the substitution models correspond to a model of the process in which the student first decides how many math courses they will take in high school and then allocates coursework within that constraint. In the substitution models, “holding all other predictors constant” includes holding the constrained total T ₁ constant and thereby incorporating the constraint in the decision process. Specifically, here the referent model tests all courses against the standard Algebra 1 and the ordered models test the level of coursework as promoted by NCES.

Overall, the test of the global hypothesis suggests the expected increase in achievement with one CU of math coursework varies as a function of the qualitative course content. All other courses were associated with a higher expected increase than Algebra 1, and the predicted ordering of regression weights was partially supported. Substantively, results must be viewed with caution for two reasons. First, Algebra 1 does not reflect taking Algebra 1 as such, but rather the effect of delaying Algebra 1 until high school. Second, a difference in the weights for two courses may be due to differences in course content or differences in the students who choose to take the different courses.

Conclusions

Methods for estimating regression weight contrasts and their standard errors exist in the literature (e.g., Graybill, 1976). The major contributions of this manuscript stem from the idea of contrasts in ANOVA (a staple) and the connection to predictors in regression. As in ANOVA, there are infinitely many possible contrasts. Also, as with ANOVA, some are more theoretically relevant. We list two such models; the referent and ordered models in which contrasts between regression weights in the original linear (or generalized linear) model are the regression weights of the reparameterized model. These models contribute in several ways. First, they show the relation between regression and corresponding referent and ordered models in ANOVA. Second, they enable the study of the impact on a criterion of qualitative differences for predictors. Third, the reparameterized models provide a tool for studying how the allocation of points from one predictor to another affects the expected value of Y holding other predictors and the total of the predictors for each person constant. The substitution models provide the estimates of contrasts, their standard errors, and tests of hypotheses about the contrasts. These models also provide a way to study moderating effects of respondent variables (e.g., sex), interactions of predictors, multicollinearity of predictors, and effects of adding/subtracting predictors. The effect of adding or subtracting a predictor from the reparameterized model generally will not be the same as that for adding or subtracting the same variable to the ordinary linear model. Also, multicollinearity of a variable will be different in the reparameterized model.

As a basis for testing hypotheses about contrasts, our approach differs from some in the literature. Consider the hypothesis $H_{0} : b_{1} = b_{2}$ . One approach is to fit two models, one in which the two weights are constrained equal and one in which they are free to vary, and then use a test statistic, such as F, to test the null hypothesis that models with and without the constraint fit equally well: for example, Davison et al. (2020) or the R software “restriktor” of Vanbrabant et al. (2022). Rather than fitting two models and applying parameter constraints, our approach involves reparameterizing the model, so that one regression weight in the reparameterized model provides an estimate of $b_{1} - b_{2}$ and its standard error which can then be used to test the null hypothesis $H_{0} : b_{1} = b_{2}$ . There are advantages and disadvantages to these differing approaches. Our approach requires no special software, provides a direct estimate of the contrast $b_{1} - b_{2}$ , requires fitting only one model, and can estimate multiple contrasts within one model. In contrast, some of the approaches involving constraints can be applicable to nonlinear constraints (Vanbrabant et al., 2022 but not Davison et al., 2020), but they may require special software and can be cumbersome in testing multiple hypotheses about multiple contrasts, because two models must be fit for each contrast. There may be times when the two approaches can be combined to complement each other. If the approach of Vanbrabant et al. (2022) is applied to our ordered model, it can be used to impose inequality constraints on contrasts: for example, $b_{2} - b_{1} < b_{4} - b_{3}$ .

Models discussed here are particularly useful for predictors measured in common units. Time is one such example. If a researcher is studying several activities, they can measure time spent in each. Regressing the criterion on the activity time predictors, a researcher can study whether the expected increase in the criterion is the same for all activities. In our example, we studied time spent in math education activities and its relationship to math achievement. Another example is cost. If a researcher is studying types of expenditures, the researcher can measure the amount spent on each expenditure. By regressing an outcome variable on the expenditure predictors, the researcher can study whether the expected increase in the outcome variable is the same for a dollar increase for each type of expenditure. A third useful unit is dosage.

Of special note, the analyses discussed here can be useful in analyzing data from an experiment or quasi-experiment. Treatments often involve multiple activities. In a study of a treatment containing three activities, the researcher can measure the amount of time spent by each respondent in each activity: for example, individual therapy, group therapy, and journaling (or the cost of each activity). By regressing the criterion on the activity predictors, one can evaluate whether a unit increase in an activity is associated with the same expected increase in outcome irrespective of activity type. The analyses can also be useful in broken randomized field trials in which respondents are randomly assigned to treatments, but some spend more time in their assigned treatment. Or some respondents not only experience their assigned treatment, but also one or more alternative treatments. These analyses give a way to analyze the experience of treatment rather than the intent to treat. In studies with a control group, the control treatment predictor can be the reference predictor in our referent model. In ANOVA, treatment is usually conceptualized as a discrete, all-or-none event, whereas in our models, a treatment can be conceptualized as an event that occurs in some amount varying over respondents and is represented in our model as a continuous predictor variable. Thus, the intensity of the treatment can also be taken into account.

In all simulation scenarios in this study, Type I error rates were adequately controlled by both the F and LM statistics. This was true for both the pairwise and global test. These results do not preclude the possibility of quite different results in other scenarios, such as that studied by Kiviet (1986) who studied time-series data to make decisions about model misspecification rather than regression weight contrasts. Our situation involves only a single time point and decisions about the equality of two or more regression coefficients.

Power was a complex function of sample size and $Δ r_{x y}$ with mean $r_{x y}$ and $r_{x x'}$ playing lesser roles. Holding sample size, $Δ r_{x y}$ , and mean $r_{x y}$ constant, power increased as a function of $r_{x x'}$ . Table 2 shows that, controlling sample size, $Δ r_{x y}$ and mean $r_{x y}$ , differences between regression weights grow as $r_{x x'}$ increases. Results were similar for the F and LM statistics. Since the t-statistic of the pairwise test is the square root of F of the global test for two predictors, the global and pairwise tests are identical for two predictors. Performance for four predictors was different, with the global test displaying greater power than the pairwise test. With sample sizes of 40 or 100 and two predictors, both tests were inadequately powered for most conditions. With sample sizes of 1,000, the tests were generally adequately powered for the effects of $Δ r_{x y}$ = .3 or .2 and sometimes for $Δ r_{x y}$ = .1 with larger mean $r_{x y}$ . With four predictors, the global test was adequately powered for $Δ r_{x y}$ = .3, or .2 for sample sizes of 40 or 100 in most conditions, with the exception of some involving low mean $r_{x y}$ and/or $r_{x x'}$ . With 500 or 1,000 observations in the sample, the tests were adequately powered for most conditions, except some with low mean $r_{x y}$ and/or $r_{x x'}$ , where power was adequate to detect only the largest effect $Δ r_{x y}$ = .3 and sometimes not even that, particularly when mean $r_{x y}$ and/or $r_{x x'}$ were low. With sample sizes of 500–1,000, the pairwise tests tended to be adequately powered for the effects of $Δ r_{x y}$ = .3, or .2, again with exceptions involving mean $r_{x y}$ and/or $r_{x x'}$ that were low. Effects of $Δ r_{x y}$ = .1 were adequately powered with sample sizes of 500–1,000 and larger values of mean $r_{x y}$ and/or $r_{x x'}$ .

There were two surprising results involving multicollinearity. Larger values of $r_{x x'}$ had virtually no effects on Type I errors (see also Lavery et al., 2019). To the extent that it had an effect on power, power increased with higher values of $r_{x x'}$ . By using proxy variables that are overlapping sums, the substitution approach can introduce higher levels of multicollinearity but multicollinearity did not have any adverse effects on Type I errors or power.

We extended the work of Carlton-Ford (1993) to V predictors. Note that our models can include regular predictors as well as contrasts. With more than two predictors, the substitution approach can have infinite manifestations, two of which were given here; the referent and ordered reparameterization. The global F-test for these models is well-known, but our new models lead to simple t tests of differences between pairs of regression coefficients in the original regression model, tests that will be of most interest for variables measured in common units. An alternative to our approach is the LM approach based on a penalty function. Rationale for the Lagrange statistic is weaker than that for the F- or t-statistic in small samples, since the former is based on asymptotic, large sample theory. Still, in our simulation study, the Lagrange performance was not markedly different from that of the F- and t-statistics.

When software for the implementation of a penalty function is readily available, the Lagrange approach can be extended to null hypotheses beyond our substitution approach by simple matrix specification of the equality restriction expressed in the null hypothesis. Substitution has the advantage that it can readily be extended beyond the ordinary least squares linear model. It can also be implemented in ridge, maximum likelihood, probit, logit, and multinomial regression. It can be implemented in virtually any software package for linear, hierarchical linear, and/or generalized linear regression. Our simulation suggests the Lagrange and substitution approaches have similar Type I error and power. Choice between the two may be a function of other factors: available software, type of regression model to be fitted (e.g., linear vs. probit), optimization function (e.g., least squares vs. ridge regression), and the form of the pairwise hypotheses.

In the ordinary regression model, regression weights are interpreted as effects of adding a unit to a predictor. In the reparametrized model, the weights correspond to contrasts between regression weights in the original model, and the contrasts are interpreted as the effect of transferring a point from one predictor to another. Our models enable the researcher to go beyond simple estimation and hypothesis testing of contrasts. The models allow one to study the effect of interactions between predictor variables in the contrast model, moderator effects of respondent variables (e.g., sex) on contrasts, multicollinearity of predictors, and the effects on R ² of adding or deleting variables. Moreover, none of this requires specialized software.

Appendix

Ordered Substitution Approach

Here, we prove that Equation 4a and b is the reparameterization of the OLS model of Equation 2a. Let t be a subscript designating a new parametrization (t = 1,…., V), where V is the number of original predictors. Each iteration results in a new parameterization of the linear regression equation, parameterization t. Let the ordinary linear regression equation (2a) be defined as parameterization t = 1.

In each iteration t, the same weighted sum of predictors is both added to and subtracted from parameterization t, which after some rearrangement of terms and a little algebra yields the outcome parameterization (t + 1). The sum that is added to and subtracted from parameterization t to obtain the parameterization (t + 1) is of the form

\sum_{v = t + 1}^{v = V} b_{t}^{t} X_{v},

where $b_{t}^{t}$ is the regression weight for predictor v in parameterization t. The iterative process is completed V − 1 times to fully reparameterize the original model in terms of regression weight contrasts. Thus, we fully describe the process for four predictors with three iterations. Extension to any number of predictors is the continuation of the same process. The generalization is also given.

Step by Step: V = 4

For four predictors, the linear regression prediction equation is as follows:

Y^{'} = b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} + a,

where X_v is predictor v, and $Y^{'} = E (Y | X_{1} X_{2} X_{3} X_{4})$ . Equation A2 is parameterization t = 1.

First Iteration involves adding and subtracting the following sum

\sum_{v = t + 1}^{v = V} b_{t}^{t} X_{v} = \sum_{v = 2}^{v = 4} b_{1} X_{v} .

If we add and subtract this sum to parameterization 1 (Equation A2), we get

Y^{'} = b_{1} X_{1} + b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} + a + \sum_{v = 2}^{v = 4} b_{1} X_{v} - \sum_{v = 2}^{v = 4} b_{1} X_{v} + a,

A3a

= (b_{1} X_{1} + \sum_{v = 2}^{v = 4} b_{1} X_{v}) + (b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} - \sum_{v = 2}^{v = 4} b_{1} X_{v}) + a,

A3b

= (b_{1} X_{1} + b_{1} X_{2} + b_{1} X_{3} + b_{1} X_{4}) + (b_{2} X_{2} + b_{3} X_{3} + b_{4} X_{4} - b_{1} X_{2} - b_{1} X_{3} - b_{1} X_{4}) + a,

A3c

= b_{1} (X_{1} + X_{2} + X_{3} + X_{4}) + (b_{2} - b_{1}) X_{2} + (b_{3} - b_{1}) X_{3} + (b_{4} - b_{1}) X_{4} + a .

A3d

If we define T_v as the sum of predictor variables v to V, then the first summation on the right of Equation A3d is $b_{1} T_{1} = b_{1} \sum_{v = 1}^{V = 4} X_{v}$ , and we can rewrite Equation A3d as parameterization 2:

Y^{'} = b_{1} T_{1} + \sum_{v = 2}^{v = 4} (b_{v} - b_{1}) X_{v} + a,

A3e

= b_{1} T_{1} + {(b}_{2} - b_{1}) X_{2} + {(b}_{3} - b_{1}) X_{3} + (b_{4} - b_{1}) X_{4} + a,

A3f

Equation A3e or A3f is the outcome parameterization t = 2, outcome for the first iteration.

Second iteration involves adding and subtracting the following sum:

\sum_{v = t + 1}^{v = V} b_{t}^{t} X_{v} = \sum_{v = 3}^{v = 4} (b_{2} - b_{1}) X_{v} .

If we add and subtract this sum to parameterization 2 (Equation A3f), we get

Y^{'} = b_{1} T_{1} + (b_{2} - b_{1}) X_{2} + {(b}_{3} - b_{1}) X_{3} + (b_{4} - b_{1}) X_{4} + a + \sum_{v = 3}^{v = 4} (b_{2} - b_{1}) X_{v} - \sum_{v = 3}^{v = 4} (b_{2} - b_{1}) X_{v},

A4a

= b_{1} T_{1} + [(b_{2} - b_{1}) X_{2} + \sum_{v = 3}^{v = 4} (b_{2} - b_{1}) X_{v}] + [(b_{3} - b_{1}) X_{3} + (b_{4} - b_{1}) X_{4} - (b_{2} - b_{1}) X_{3} - (b_{2} - b_{1}) X_{4}] + a,

A4b

= b_{1} T_{1} + [(b_{2} - b_{1}) X_{2} + (b_{2} - b_{1}) X_{3} + (b_{2} - b_{1}) X_{4})] + [(b_{3} - b_{1}) X_{3} + (b_{4} - b_{1}) X_{4} - (b_{2} - b_{1}) X_{3} - (b_{2} - b_{1}) X_{4}] + a,

A4c

= b_{1} T_{1} + (b_{2} - b_{1}) (X_{2} + X_{3} + X_{4}) + (b_{3} - b_{1} - b_{2} + b_{1}) X_{3} + (b_{4} - b_{1} - b_{2} + b_{1}) X_{4} + a .

A4d

Substituting $T_{2} = \sum_{v = 2}^{v = 4} X_{v}$ into the previous equation and simplifying gives

= b_{1} T_{1} + (b_{2} - b_{1}) T_{2} + (b_{3} - b_{2}) X_{3} + (b_{4} - b_{2}) X_{4} + a .

A4e

Equation A4e is parameterization t = 3. We are now ready for the last iteration.

Third iteration. For this iteration, we need to add to and subtract from parameterization 3 (Equation A4d) the sum

\sum_{v = t + 1}^{v = V} b_{t}^{t} X_{v} = (b_{3} - b_{2}) X_{4} .

Adding and subtracting this quantity from parameterization 3 (Equation A4d) yields

Y^{'} = b_{1} T_{1} + (b_{2} - b_{1}) T_{2} + (b_{3} - b_{2}) X_{3} + (b_{4} - b_{2}) X_{4} + (b_{3} - b_{2}) X_{4} - (b_{3} - b_{2}) X_{4} + a,

A5a

= b_{1} T_{1} + (b_{2} - b_{1}) T_{2} + [(b_{3} - b_{2}) X_{3} + (b_{3} - b_{2}) X_{4}] + [(b_{4} - b_{2} - b_{3} + b_{2}) X_{4}] + a .

A5b

Substituting $T_{3} = \sum_{v = 3}^{v = 4} X_{v}$ = $X_{3} + X_{4}$ into the previous equation and simplifying yields

Y^{'} = b_{1} T_{1} + (b_{2} - b_{1}) T_{2} + (b_{3} - b_{2}) T_{3} + (b_{4} - b_{3}) X_{4} + a .

A5c

Equation A5c is parameterization t = 4 and the final one. It is a regression equation in which three of the predictors are sums ( $T_{1}, T_{2}, and T_{3})$ and the last is the predictor X ₄. The regression weights on three of the predictors ( $T_{2}, T_{3}, and X_{4})$ are contrasts between adjacent regression weights in the original model of Equation A2. If we regress Y onto the four predictors $(T_{1}, T_{2}, T_{3}, and X_{4})$ , then the familiar test of the null hypothesis that the regression weights equal zero will yield tests of the following four hypotheses: $H_{0} : b_{1} = 0$ , $H_{0} : b_{2} - b_{1} = 0$ , $H_{0} : b_{3} - b_{2} = 0$ , and $H_{0} : b_{4} - b_{3} = 0$ , relative to the original Equation A2.

Equation A5c gives the final parameterization for the special case of four predictors. For any number of predictor variables V, the general form for the final reparameterization is

Y^{'} = b_{1} T_{1} + \sum_{v = 2}^{v = V - 1} (b_{v} - b_{v - 1}) T_{v} + (b_{V} - b_{V - 1}) X_{V} + a .

If the predictors ( $T_{1} \dots, T_{V - 1}, X_{V})$ are regressed onto Y, the familiar t-statistic for testing the null hypothesis that regression weights equal 0.0 yields tests of the null hypothesis $H_{0} : b_{1} = 0$ and the null hypotheses that $H_{0} : b_{v} - b_{v - 1} = 0$ for all v = 2,…, V. Again, relative to the original regression equation (A2).

Supplemental Material

Supplemental Material, sj-docx-1-jeb-10.3102_10769986231200155 - Pairwise Regression Weight Contrasts: Models for Allocating Psychological Resources

Supplemental Material, sj-docx-1-jeb-10.3102_10769986231200155 for Pairwise Regression Weight Contrasts: Models for Allocating Psychological Resources by Mark L. Davison, Hao Jia and Ernest C. Davenport in Journal of Educational and Behavioral Statistics

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Bradbury

N. A.

(2016). Attention span during lectures: 8 seconds, 10 minutes, or more? Advances in Physiology Education, 40(4), 509–513. https://doi.org/10.1152/advan.00109.2016

Bradley

J. V.

(1978). Robustness? British Journal of Mathematical and Statistical Psychology, 31(2), 144–152. https://doi.org/10.1111/j.2044-8317.1978.tb00581.x

Buse

(1982). The likelihood ratio, Wald, and Lagrange multiplier tests: An expository note. The American Statistician, 36(3), 153–157. https://doi.org/10.2307/2683166

Carlton-Ford

(1993). A simple method for estimating and testing equality restrictions using OLS. Sociological Focus, 26(2), 165–176. https://doi.org/10.1080/00380237.1993.10571004

Cooper

Atkinson

E. J.

Wahner

H. W.

O’Fallon

W. M.

Riggs

B. S.

Judd

H. L.

Melton

L. J., III.

(1992). Is caffeine consumption a risk factor for osteoporosis? Journal of Bone Mineral Research, 7(4), 465–471. https://doi.org/10.1002/jbmr.5650070415

Cowan

(2005). Working memory capacity (1st ed.). Psychology Press. https://doi.org/10.4324/9780203342398

Davison

M. L.

Chang

Y.-F.

Davenport

E. C.

Jr. (2014). Modeling configural patterns in latent variable profiles: Association with an endogenous variable. Structural Equation Modeling, 21, 81–93. https://doi.org/10.1080/10705511.2014.859507

Davison

M. L.

Davenport

E. C.

Hao

(2023). Linear equality constraints: Reformulations of criterion related profile analysis with extensions to moderated regression for multiple groups. Psychological Methods, 28(3), 600–612. https://doi.org/10.1037/met0000430

Davison

M. L.

Davenport

E. C.

Jr. Kohli

Kang

Park

(2020). Addressing quantitative and qualitative hypotheses using regression models with equality restrictions and predictors measured in common units. Multivariate Behavioral Research, 56(1), 86–100. https://doi.org/10.1080/00273171.2020.1754154

10.

Davison

M. L.

Jew

Davenport

E. C.

Jr. (2014). Patterns of SAT Scores, choice of STEM major, and gender. Measurement and Evaluation in Counseling and Development, 47, 118–126. https://doi.org/10.1177/0748175614522269

11.

Engle

R. F.

(1984). Wald, likelihood ratio, and Lagrange multiplier tests in econometrics. Handbook of Econometrics, 2, 775–826. https://doi.org/10.1016/S1573-4412(84)02005-5

12.

Evans

G. B. A.

Savin

N. E.

(1982). Conflict among the criteria revisited; the W, LR and LM tests. Econometrica: Journal of the Econometric Society, 50(3), 737–748. https://doi.org/10.2307/1912611

13.

Graybill

F. A.

(1976). Theory and application of the linear model. Duxbury Press.

14.

Ingels

S. J.

Pratt

D. J.

Herget

Bryan

Fritch

L. B.

Ottem

Rogers

J. E.

Wilson

(2015). High School Longitudinal Study 2009, 2013 (HLS 09) Update and High School Transcript Data File Documentation (NCES 2015-036). National Center for Education Statistics, Institute of Education Sciences, U. S. Department of Education: Washington, DC. http://nces.ed.gov/pubsearch

15.

Ingels

S. J.

Pratt

D. J.

Herget

D. R.

Dever

J. A.

Fritch

L. B.

Ottem

Rogers

J. E.

(2014). High School Longitudinal Study of 2009 (HSLS:09) Base Year to First Follow-Up Data File Documentation (NCES 2014-361). https://nces.ed.gov/pubs2014/2014361.pdf

16.

Kish

Frankel

M. R.

(1974). Inference from complex samples. Journal of the Royal Statistical Society, 36(1), 1–37. https://doi.org/10.1111/j.2517-6161.1974.tb00981.x

17.

Kiviet

J. F.

(1986). On the rigour of some misspecification tests for modelling dynamic relationships. The Review of Economic Studies, 53(2), 241–261. https://doi.org/10.2307/2297649.

18.

Lavery

M. R.

Acharya

Sivo

S. A.

(2019). Number of predictors and multicollinearity: What are their effects on error and bias in regression?. Communications in Statistics-Simulation and Computation, 48(1), 27–38. https://doi.org/10.1080/03610918.2017.1371750

19.

Lee

V. E.

Burkham

D. T.

Chow-Roy

Smerdon

B. A.

Geverdt

(1998). High school curriculum structure: Effects on course-taking and achievement in mathematics for high school graduates. (NCES Publication No. 98-09). National Center for Education Statistics. Washington, DC: U.S. Department of Education. https://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=9809

20.

McClelland

M. M.

Acock

A. C.

Piccinin

Rhea

S. A.

Stallings

M. C.

(2013). Relations between preschool attention span-persistence and age 25 educational outcomes. Early Childhood Research Quarterly, 28(2), 314–324. https://doi.org/10.1016/j.ecresq.2012.07.008

21.

Oberauer

Farrell

Jarrold

Lewandowsky

(2016). What limits working memory capacity? Psychological Bulletin, 142(7), 758–789. https://doi.org/10.1037/bul0000046

22.

Rindskopf

(1984). Linear equality restrictions in regression and loglinear models. Psychological Bulletin, 96, 597–603. https://doi.org/10.1037/0033-2909.96.3.597

23.

Schnitker

S. A.

(2012). An examination of patience and well-being. The Journal of Positive Psychology, 7(4), 263–280. https://doi.org/10.1080/17439760.2012.697185

24.

Vanbrabant

Rosseel

Dacko

(2022). Restriktor: Restricted Statistical Estimation and Inference for Linear Models. R package version 0.4-500. https://CRAN.R-project.org/package=restriktor

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