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This paper examined the amount bias in standard errors for fixed effects when the random part of a multilevel model is misspecified. Study 1 examined the effects of misspecification for a model with one Level 1 predictor. Results indicated that misspecifying random slope variance as fixed had a moderate effect size on the standard errors of the fixed effects and had a greater effect than misspecifying fixed slopes as random. In Study 2, a second Level 1 predictor was added and allowed for the examination of the effects of misspecifying the slope variance of one predictor on the standard errors for the fixed effects of the other predictor. Results indicated that only the standard errors of coefficient relevant to that predictor were impacted and that the effect size for the bias could be considered moderate to large. These results suggest that researchers can use a piecemeal approach to testing multilevel models with random effects.

The use of multilevel modeling has become common in social science research because it allows researchers to analyze data that reside at different levels. Multiple levels may include employees nested within teams, students nested within classrooms, or measurements nested within individuals. A strength of multilevel modeling is the flexibility that it allows researchers in testing how variables at higher levels relate to variables at lower levels. A cost of the flexibility is that multilevel models can be complicated to specify. As such, studies have investigated the effects of misspecifying multilevel models. For example, studies have investigated the effects of ignoring (a) the heterogeneity of Level 2 variances (

In multilevel modeling, specifying the random part of the model is often the most difficult because there are rarely strong theories about which effects should vary. Proper specification of the random part of the model is important because it influences the standard errors of the fixed effects (

Despite its importance, there has been little research concerning the effects of misspecifying the random part of the model. One exception is

In the present studies, we examined the consequences of misspecifying the random part of the model on the standard errors of the fixed effects. We were interested in the effects of specifying fixed slopes as varying and varying slopes as fixed. Because closed-form expressions are not possible when there are unequal group sizes or random slopes, we used Monte Carlo simulations to assess the effects of model misspecification. In our simulations, we assessed bias by comparing standard errors for the fixed effects from a model where the slope varied to a model where the slope was fixed. Our goal was to determine which type of misspecification produced the most detrimental effects on the standard errors of the fixed effects.

A succinct way of expressing a basic two-level model where there are

In _{ij}^{2}) represents within group variance not explained by the model. The subscripts _{0j} is represented as τ_{00} and represents intercept variance not explained by the _{hj}_{hh} and represent the unexplained variances in the Level 1 slopes. It is common to allow the intercept and slopes to covary.

Researchers are often advised to test for the significance of the variance components in order to determine what effects should vary. One potential problem with using significance tests for the slope variance components is that the power for these tests is low (

To understand how the misspecification of the random part of the model affects the standard errors for the fixed effects, it is helpful to consider the unrealistic case where the variance components are known, every Level 1 predictor has a random slope, and each group has sufficient size to calculate ordinary least squares (OLS) estimates. Following

where _{j}_{i}_{j}_{i}_{j}_{j}^{2}. The ordinary least squares (OLS) estimator for the unknown parameters is

The error variance matrix for _{j}_{j}

Finally, pre-multiplying both sides of

At Level 2, the model for β_{j}

In _{j}_{j}

Both _{0j} and _{1j} are the random effects representing the intercept and slope, respectively. They are assumed to be normally distributed with a mean vector of zeros and a variance covariance matrix

Substituting

The dispersion of _{j}

Finally, the dispersion matrix for the fixed effects (γ’s) based on generalized least squares is given by:

The square roots of the diagonals of _{γ} are the standard errors. To trace the effects of Δ_{j} on the standard errors, it is helpful to consider a very simple model with two Level 1 predictors and no Level 2 predictors. The underlying concepts are identical for models with more fixed and random effects but are more difficult to demonstrate. Thus, in our simple model,

standard errors for γ_{00}, γ_{10,} and γ_{20} are simply the square roots of the diagonals of _{j}

In practice, the variance components are not known and need to be estimated with the fixed effects using full or restricted maximum likelihood estimation. Based on the above discussion, we would expect the bias in standard errors would be limited to those fixed effects that are associated with the random effect that is misspecified and that the amount of this bias should decrease as the amount of slope variance decreases. In addition, the amount of bias in standard errors should increase as group size increases. This is because, all things being equal, the elements of the error matrix _{j}

The major focus of Study 1 was to examine the effects of omitting a slope from the random part of the model. We also explored the effects of freeing a slope when it should have been fixed. We considered a simple model with only one Level 1 predictor and one Level 2 predictor. As mentioned above, specification of the random part of the model impacts the standard errors for the fixed effects. We assessed the effect of misspecification on the standard errors of three fixed effects: the cross-level effect (γ_{01}), the slope mean for _{10}), and the cross-level interaction (γ_{11}). Misspecification occurred by allowing the slope to vary when it should have been fixed and vice-versa. For each fixed effect, we were interested in the overall degree of bias. In addition, we assessed the impact of the number of groups, average group size, the mean slope, the magnitude of the cross-level interaction, and the amount of slope variance on the amount of bias.

We used a 3 (number of groups) × 3 (average group size) × 2 (cross-level interaction effect size) × 4 (slope mean) × 3 (slope – intercept correlation) × 5 (slope variance effect size) design. The values chosen for the conditions were based somewhat on the simulations conducted by

We created the Level 2 variable _{0j}, the standard deviation was set to the value to produce an ICC for _{1j}, the standard deviation was set to the value to produce the desired slope variance. At Level 1, we randomly sampled values from a standard normal distribution to produce the Level 1 predictor _{ij}

To generate the _{0j}. Similarly, we created the group slopes (β_{1j}) by adding the desired mean slope (i.e., 0.00, 0.10, 0.30, or 0.50) and the product of _{1j}. Finally, we computed values for _{ij}

For each sample, we estimated two multilevel models using restricted maximum-likelihood estimation: one where the slope was fixed and one where the slope varied. For each model, we recorded the estimates for γ_{01}, γ_{10}, and γ_{11}, as well as their standard errors. We calculated the bias in standard errors by subtracting the standard error from the data generating model from that of the misspecified model. Thus, a positive value for bias reflects the standard errors for the misspecified model was higher than the data generating model. Cohen’s ^{2}). All analyses were conducted using the LME4 (

Prior to evaluating the results, we inspected how well the population parameters were uncovered by our analyses. We calculated root mean square error of approximation values (RMSEA) for each parameter. RMSEA values ranged from .02 to .03. Thus, all values were in the acceptable range and consistent with prior simulations.

_{01}. The Cohen’s

Source | γ_{01} |
γ_{10} |
γ_{11} |
||||||
---|---|---|---|---|---|---|---|---|---|

NJ = 5 | .090 | .000 | 0.01 | .061 | -.009 | -0.52 | .062 | -.009 | -0.47 |

NJ = 10 | .079 | .000 | 0.01 | .046 | -.010 | -0.77 | .046 | -.010 | -0.72 |

NJ = 20 | .075 | .000 | 0.01 | .037 | -.012 | -1.14 | .037 | -.012 | -1.08 |

NG = 50 | .110 | .000 | 0.02 | .066 | -.014 | -0.84 | .067 | -.014 | -0.76 |

NG = 100 | .078 | .000 | 0.02 | .046 | -.010 | -0.87 | .046 | -.010 | -0.82 |

NG = 200 | .055 | .000 | 0.02 | .032 | -.007 | -0.88 | .033 | -.007 | -0.85 |

γ_{10} = 0.00 |
.081 | .000 | 0.01 | .048 | -.010 | -0.59 | .049 | -.011 | -0.56 |

γ_{10} = 0.10 |
.081 | .000 | 0.01 | .048 | -.010 | -0.59 | .049 | -.011 | -0.56 |

γ_{10} = 0.30 |
.081 | .000 | 0.01 | .048 | -.010 | -0.59 | .049 | -.010 | -0.56 |

γ_{10} = 0.50 |
.081 | .000 | 0.01 | .048 | -.010 | -0.59 | .049 | -.010 | -0.56 |

τ_{11} = 0.00 |
.044 | .000 | -0.02 | .035 | .001 | 0.09 | .036 | .001 | 0.09 |

τ_{11} = 0.025 |
.084 | .000 | 0.00 | .041 | -.005 | -0.28 | .042 | -.005 | -0.26 |

τ_{11} = 0.05 |
.084 | .000 | 0.01 | .045 | -.008 | -0.47 | .046 | -.008 | -0.44 |

τ_{11} = 0.10 |
.084 | .000 | 0.01 | .052 | -.014 | -0.78 | .052 | -.014 | -0.74 |

τ_{11} = 0.15 |
.084 | .001 | 0.02 | .058 | -.019 | -1.03 | .058 | -.019 | -0.96 |

τ_{01} = 0.00 |
.076 | .000 | 0.01 | .047 | -.009 | -0.53 | .047 | -.009 | -0.50 |

τ_{01} = -0.30 |
.084 | .000 | 0.01 | .049 | -.011 | -0.64 | .050 | -.011 | -0.60 |

τ_{01} = -0.70 |
.084 | .000 | 0.01 | .049 | -.011 | -0.62 | .049 | -.011 | -0.59 |

γ_{11} = 0.00 |
.081 | .000 | 0.01 | .048 | -.010 | -0.59 | .049 | -.011 | -0.56 |

γ_{11} = 0.20 |
.081 | .000 | 0.01 | .048 | -.010 | -0.59 | .049 | -.010 | -0.56 |

Misspecification of the random part of the model did influence the standard errors for γ_{10} and γ_{11}. Cohen’s d’s ranged from -1.14 to 0.09 for γ_{10} and -1.08 to 0.09 to for γ_{11.} The smallest values were found for the no slope variance condition, and the biggest values was found for the largest group size condition. When there was no population slope variance, estimating a model with a random slope produced a bias of .001 (Cohen’s _{10} and γ_{11}. In contrast, misspecifying the slope as fixed when it should have been random produced biases ranging from -.019 to -.005 for both γ_{10} and γ_{11}. Cohen’s _{10} and -0.96 to -0.26 for γ_{11}. As expected, bigger biases occurred for greater levels of slope variance.

To get a better sense of the effects of the study conditions on bias, we conducted separate ANOVA’s investigating the impact of the study characteristics on the bias in standard errors for the γ_{10} and γ_{11}. We included only the conditions where there was slope variance to produce a fully crossed design. ^{2} = .65 for γ_{10} and .62 for γ_{11}). The number of groups (partial η^{2} = .38 for γ_{10} and .36 for γ_{11}) also had a sizable effect size. The partial η^{2} for group size was .11 for γ_{10} and .10 for γ_{11}. Based on ^{2}s = .00). Finally, only the interaction between the number of groups and amount of slope variance had a sizeable effect (partial η^{2}s = .11). _{10}. The effect for slope variance decreased as the number of groups increased.

Source | Bias in γ_{10} |
Bias in γ_{11} |
|||
---|---|---|---|---|---|

Partial η^{2} |
Partial η^{2} |
||||

Number of cases per group (NJ) | 2 | 54013.35* | .11 | 50272.89* | .10 |

Number of groups (NG) | 2 | 264087.34* | .38 | 240527.90* | .36 |

Slope Mean (SM) | 3 | 0.54 | .00 | 0.66 | .00 |

Slope variance (SV) | 3 | 521630.03* | .65 | 466481.81* | .62 |

Cross-level interaction (CINT) | 1 | 2.51 | .00 | 3.86* | .00 |

Intercept – slope correlation | 2 | 660.27* | .00 | 576.28* | .00 |

NG × SV | 6 | 18279.90* | .11 | 16739.40 | .10 |

*

Absolute bias of the standard errors for _{10} as a function of the amount of slope variance and number of groups.

In Study 2, we assessed the effects of misspecifying the random part of the model when there were multiple random slopes by adding a second Level 1 predictor. There is potential for greater effects as this also influences the correlations between that slope variance with the variances of the intercept and other slopes. We were interested in how misspecifying slope variance for _{1} impacted the standard errors for the slope mean for _{2} (γ_{20}) and how misspecifying slope variance for _{2} affected the standard errors for the slope mean for _{1} (γ_{10}) and the cross-level interaction (γ_{11}).

We used a 3 (number of groups) × 3 (average group size) × 4 (slope correlation) × 5 (slope variance effect size for _{1}) × 2 (slope variance effect size for _{2}) × 4 (predictor correlation) × 2 (between group variance for _{1}) × 2 (between group variance for _{2}) design. It was not fully crossed because there can be no slope correlation when either of the slopes have zero variance. The values for the number of groups, average group size, and slope variance effect size for _{1} were consistent with Study 1. The cross-level interaction and slope mean of _{1} were set at 0.20 and 0.30 respectively. We eliminated the other conditions for these variables because Study 1 demonstrated negligible differences in standard error bias across these conditions. In addition, we added another Level 1 predictor (_{2}). We set the correlation between the two Level 1 predictors (_{X1X2}) to be 0.00, 0.50, and 0.90. The slope mean for _{2} was also set to 0.30, and the slope variance was set to either 0.00 or 0.10. We set the correlation between the two slopes to be .00, .50, or .90. When the slope variance for _{1} or _{2} was 0.00, the correlation between slopes was fixed to zero. Since Level 1 predictors can also contain between group variance, we set the ICC for each predictor to be .00 or .50. The latter value represents a substantial ICC value. For each possible combination, we simulated 1000 samples resulting in 3,168,000 samples.

The Level 2 variables _{0j} were generated as in Study 1. For _{1j}, the standard deviation was set to the value to produce the desired slope variance. For _{2j}, the standard deviation was set to the value of either 0.00 or the square root of 0.10 depending on the condition. We transformed these values to produce the desired covariance matrix for random effects using procedures adapted from nonparametric bootstrapping (_{1} and _{2}, we randomly sampled values from a standard normal distribution to produce the desired within group variance. Between group variance for _{1} and _{2} was created by adding the desired between group variances to the _{1} and _{2} predictors. In addition, the Level 1 error term, _{ij}

As in Study 1, we estimated multiple models. In Model 1, the slope for _{1} was fixed and the slope for _{2} was random. In Model 2, the _{1} slope was random and the _{2} slope was fixed. Finally, Model 3 specified both slopes as random. We compared the Model 1 standard errors for the fixed effects against those from Model 3 to determine the effects of misspecifying the slope of _{1} on the standard errors of γ_{01}, γ_{10}, γ_{11}, and γ_{20}. We also compared Model 2 with Model 3 to determine if misspecifying the slope for _{2} affected the standard errors for γ_{11}.

_{01}_{,}_{10}_{,}_{11} were similar to the one-predictor case. Misspecifying the slope of _{1} had little effect on the standard errors of γ_{01} with bias estimates ranging from .000 to .002 and Cohen’s d’s ranging from 0.01 to 0.07. There was considerable bias in the standard errors for both γ_{10} and γ_{11} when the slope of _{1} was incorrectly fixed to zero. Bias estimates ranged from -.016 to -.005 with Cohen’s _{10} standard errors. Similarly, bias estimates for the γ_{11} standard errors ranged from -.004 to -.016 with Cohen’s _{11} as random when it should have been fixed to zero were not as pronounced. Bias estimates for standard errors were .002 for γ_{10} and .001 for γ_{11}. Cohen’s

Source | γ_{01} |
γ_{10} |
γ_{11} |
γ_{20} |
||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

NJ = 5 | .076 | .001 | 0.03 | .082 | -.010 | -0.26 | .067 | -.008 | -0.36 | .081 | .002 | 0.06 |

NJ = 10 | .064 | .001 | 0.03 | .060 | -.009 | -0.35 | .050 | -.008 | -0.51 | .059 | .003 | 0.10 |

NJ = 20 | .059 | .001 | 0.03 | .048 | -.010 | -0.47 | .042 | -.010 | -0.63 | .049 | .004 | 0.18 |

NG = 50 | .090 | .001 | 0.06 | .088 | -.013 | -0.38 | .074 | -.012 | -0.59 | .087 | .004 | 0.12 |

NG = 100 | .064 | .001 | 0.07 | .060 | -.009 | -0.36 | .050 | -.008 | -0.60 | .060 | .003 | 0.11 |

NG = 200 | .045 | .000 | 0.07 | .042 | -.006 | -0.35 | .035 | -.006 | -0.62 | .042 | .002 | 0.12 |

τ_{11} = .000 |
.066 | .000 | -0.01 | .051 | .002 | 0.08 | .040 | .001 | 0.07 | .058 | .001 | 0.02 |

τ_{11} = .025 |
.066 | .000 | 0.01 | .058 | -.005 | -0.16 | .047 | -.004 | -0.21 | .063 | .001 | 0.04 |

τ_{11} = .050 |
.066 | .000 | 0.02 | .061 | -.008 | -0.24 | .050 | -.007 | -0.34 | .063 | .002 | 0.07 |

τ_{11} = .10 |
.067 | .001 | 0.04 | .067 | -.012 | -0.38 | .057 | -.011 | -0.54 | .063 | .004 | 0.13 |

τ_{11} = .15 |
.067 | .002 | 0.06 | .071 | -.016 | -0.51 | .062 | -.016 | -0.70 | .064 | .006 | 0.18 |

τ_{22} = .00 |
.066 | .001 | 0.03 | .061 | -.010 | -0.28 | .048 | -.008 | -0.40 | .053 | .001 | 0.03 |

τ_{22} = .10 |
.067 | .001 | 0.03 | .064 | -.010 | -0.30 | .054 | -.009 | -0.42 | .066 | .004 | 0.11 |

τ_{12} = .00 |
.066 | .001 | 0.03 | .062 | -.009 | -0.28 | .050 | -.008 | -0.38 | .059 | .001 | 0.04 |

τ_{12} = .10 |
.067 | .001 | 0.03 | .065 | -.010 | -0.32 | .054 | -.009 | -0.45 | .066 | .003 | 0.08 |

τ_{12} = .50 |
.067 | .001 | 0.03 | .065 | -.010 | -0.31 | .056 | -.010 | -0.46 | .066 | .005 | 0.14 |

τ_{12} = .90 |
.067 | .001 | 0.04 | .065 | -.010 | -0.32 | .056 | -.010 | -0.42 | .066 | .007 | 0.19 |

ICC_{X1} = .00 |
.066 | .000 | 0.02 | .062 | -.010 | -0.30 | .052 | -.009 | -0.43 | .063 | .003 | 0.10 |

ICC_{X1} = .50 |
.067 | .001 | 0.04 | .064 | -.009 | -0.29 | .054 | -.009 | -0.40 | .063 | .003 | 0.09 |

ICC_{X2} = .00 |
.066 | .001 | 0.03 | .063 | -.010 | -0.31 | .053 | -.009 | -0.42 | .061 | .003 | 0.10 |

ICC_{X2} = .50 |
.067 | .001 | 0.03 | .064 | -.010 | -0.29 | .053 | -.009 | -0.41 | .064 | .003 | 0.09 |

r_{X1X2} = .00 |
.068 | .001 | 0.03 | .052 | -.010 | -0.58 | .051 | -.010 | -0.49 | .052 | .001 | 0.06 |

r_{X1X2} = .10 |
.067 | .001 | 0.03 | .056 | -.010 | -0.40 | .051 | -.009 | -0.45 | .056 | .002 | 0.07 |

r_{X1X2} = .50 |
.066 | .001 | 0.03 | .056 | -.010 | -0.47 | .052 | -.010 | -0.50 | .055 | .005 | 0.21 |

r_{X1X2} = .90 |
.065 | .001 | 0.03 | .090 | -.009 | -0.20 | .058 | -.007 | -0.27 | .089 | .005 | 0.10 |

_{01}, γ_{10}, γ_{11}, and γ_{20} standard errors were calculated by subtracting those from those from a model with a random _{1} slope (Model 2) from these standard errors from a model with a fixed _{1} slope (Model 1).

It is interesting to note that varying the amount of slope variance produced little bias in the standard errors for γ_{20}. In general, the amount of bias was similar to that for the standard errors of γ_{01}. In addition, the degree of correlation between slopes did not appear to impact the bias in the standard errors for γ_{10} and γ_{11}. For example, when the correlation was .00, the bias in the standard errors for γ_{10} was -.009 with a Cohen’s _{10} were -.010 with a Cohen’s _{1}, and -.009 with a Cohen’s _{11}. The between group variance conditions for _{2} produced biases and effect sizes that mimicked the findings for _{1}.

We conducted two ANOVA’s investigating the impact of the study characteristics on the bias in standard errors for the average γ_{10} and γ_{11}. As with Study 1, only conditions where there was variance in the slopes were included. We limited the ANOVA to the main effects and all two-way interactions because of the large number of conditions. ^{2} = .43 for γ_{10} and .49 for γ_{11}) and the number of groups (partial η^{2} = .27 for both γ_{10} and γ_{11}). None of the two-way interactions a partial η^{2}s greater than .06.

Source | Bias in γ_{10} |
Bias in γ_{11} |
|||
---|---|---|---|---|---|

Partial η^{2} |
Partial η^{2} |
||||

Number of cases per group | 2 | 9961.09 | 0.01 | 50982.95 | 0.04 |

Number of groups | 2 | 427913.47 | 0.27 | 436516.19 | 0.27 |

Predictor correlation | 3 | 23328.24 | 0.03 | 91597.76 | 0.11 |

Slopes correlation | 3 | 121.52 | 0.00 | 3532.56 | 0.00 |

Slope variance for X1 | 3 | 573301.99 | 0.43 | 745638.19 | 0.49 |

ICC for X1 | 1 | 1746.76 | 0.00 | 2459.09 | 0.00 |

ICC for X2 | 1 | 118.75 | 0.00 | 299.74 | 0.00 |

^{2} < .02. ICC = Intraclass correlation coefficient.

Varying the slope variance for _{2} (τ_{22}) when it should have been fixed did not affect the standard errors for γ_{10} or γ_{11}. The bias estimates were -.001 and .000 with a Cohen’s _{10} and γ_{11,} respectively. Similarly, fixing the slope variance for _{2} (τ_{22}) when it should have been random also did not have a large effect. The bias estimates were .004 and .003 with a Cohen’s _{10} and γ_{11,} respectively. This provides further evidence that the effects of misspecifying the random part of multilevel models are isolated to the standard errors for the fixed effects associated with the misspecified slope variance.

It is well known that misspecification of the random part of multilevel models impacts the standard errors for the fixed effects. In the present studies, we sought to identify how great this impact is, and what type of misspecification has the larger effect. In Study 1, we found that fixing the slope variance when it should be random has bigger effects than freeing the slope variance when it should be fixed. Large effect sizes for both the Level 1 slope and cross-level interaction standard errors were observed for relatively small levels of slope variance. The bias was such that Type I errors were more likely when the slope was fixed but should have varied, and Type II errors were more likely when the slope varied but should have been fixed. We also found that the number of groups attenuated the biasing effect of slope variance magnitude such that more groups were associated with less of a biasing effect.

In Study 2, we investigated the effects of misspecifying the random part of the model in situations with multiple random slopes and between-group variance for correlated Level 1 predictors. Overall, the results were similar to the single random slope models in that these added conditions did not have a large impact. As expected, the bias in standard errors is limited to the fixed effects associated with the slope that should be random. That is, the only fixed effects that are affected are the direct effect of the Level 1 predictor, and the cross-level interaction of the Level 2 variable predicting the Level 1 slope.

In our simulations, we considered a basic two-level model with relatively few continuous predictors. Thus, it is not clear how well the results would generalize to studies with a larger number of predictors, additional levels, or categorical variables. Similarly, we considered a model with a simple compound symmetry variance/covariance matrix for the Level 1 errors. The assumption of compound symmetry is often relaxed for longitudinal models because of autocorrelation and heterogeneity of variance that occurs for repeated measures. This directly effects the standard errors for Level 1 coefficients (

We see three major recommendations emerging from the results of the present studies. First, we suggest that researchers, when in doubt, give preference to random slopes. Our results suggest that doing so would minimize the amount of bias in the relevant standard errors for the fixed effects. We suspect that it is rare that the variance for a slope is exactly zero. However, even if this were to be the case, our results suggest that allowing that slope to vary would produce very little bias. In contrast, fixing a slope that should be random produces considerably more bias. If researchers desire to gauge the potential effects of misspecifying slope variance, then they could compare models with and without random slopes.

Second, we suggest that researchers should avoid using nonrandomly varying slope models to evaluate cross-level interactions. Cross-level interactions may exist when the variance in slopes is not statistically significant (

Finally, we recommend using a piecemeal approach when there are many random effects that may cause estimation problems or if there are not enough degrees of freedom to estimate all of the random effects. Results from Study 2 suggest that omitting a slope from the random part of the model only affects coefficients concerning the omitted slope. Researchers could test their hypotheses regarding fixed effects allowing for the appropriate random effects. For example, we may hypothesize that the Level 2 variable, Z, predicts the slopes of _{1} and _{2}. We could first examine the effects of _{1} slope while allowing the _{1} slope to vary and fixing the _{2} slope to zero. Next, we would examine the effects of _{2} slope while allowing the _{2} slope to vary and fixing the _{1} slope. Doing so would allow us to examine the hypotheses using the appropriate standard errors.

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no support to report.