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Beta coefficients for linear regression models represent the ideal form of an interpretable feature effect. However, for non-linear models such as generalized linear models, the estimated coefficients cannot be interpreted as a direct feature effect on the predicted outcome. Hence, marginal effects are typically used as approximations for feature effects, either as derivatives of the prediction function or forward differences in prediction due to changes in feature values. While marginal effects are commonly used in many scientific fields, they have not yet been adopted as a general model-agnostic interpretation method for machine learning models. This may stem from the ambiguity surrounding marginal effects and their inability to deal with the non-linearities found in black box models. We introduce a unified definition of forward marginal effects (FMEs) that includes univariate and multivariate, as well as continuous, categorical, and mixed-type features. To account for the non-linearity of prediction functions, we introduce a non-linearity measure for FMEs. Furthermore, we argue against summarizing feature effects of a non-linear prediction function in a single metric such as the average marginal effect. Instead, we propose to average homogeneous FMEs within population subgroups, which serve as conditional feature effect estimates.

This paper develops methods for evaluating marginal policy changes. We characterize how the effects of marginal policy changes depend on the direction of the policy change, and show that marginal policy effects are fundamentally easier to identify and to estimate than conventional treatment parameters. We develop the connection between marginal policy effects and the average effect of treatment for persons on the margin of indifference between participation in treatment and nonparticipation, and use this connection to analyze both parameters. We apply our analysis to estimate the effect of marginal changes in tuition on the return to going to college.

Background In regional wheat trials, when short-stem wheat varieties and high-stem wheat varieties are planted adjacent to each other in small plots, changes in their marginal plot environment can lead to bias in yield evaluation. Currently, there is no relevant research revealing the degree of their mutual influence. Results In a regional wheat experiment, when high-stem wheat varieties and short-stem wheat varieties were planted adjacent to one another, there was no significant change in soil temperature or humidity in the high-stem wheat variety experimental plot from November to May compared to the control plot, while the soil humidity in the short-stem wheat variety experimental plot was greater than that in the control plot. In May, the soil temperature of the short-stem wheat varieties in the experimental plot was lower than that in the control plot. Illumination of the wheat canopy in the high-stem wheat variety experimental plot had a significant positive effect in April and May, while illumination of the wheat canopy in the short-stem wheat variety experimental plot had a negative effect. The chlorophyll fluorescence parameters of flag leaves in the high-stem wheat variety experimental plots showed an overall increasing trend, while the chlorophyll fluorescence parameters of flag leaves in the experimental plots of short-stem wheat varieties showed a decreasing trend. The analysis of the economic yield, biological yield, and yield factors in each experimental plot revealed that the marginal effects of the economic yield and 1000-grain weight were particularly significant and manifested as positive effects in the high-stem wheat variety experimental plot and as negative effects in the short-stem wheat variety experimental plot. The economic yield of the high-stem wheat variety experimental plot was significantly greater than that of the control plot, the economic yield of the short-stem wheat variety experimental plot was significantly lower than that of the control plot, and the economic yield of the high-stem experimental plot was significantly greater than that of the short-stem experimental plot. When the yield of the control plot of the high-stem wheat varieties was compared to that of the control plot of the short-stem wheat varieties, the yield of the control plot of the short-stem wheat varieties was significantly greater than that of the control plot of the high-stem wheat varieties. Conclusions Based on these findings, it is concluded that plots with high-stem and short-stem wheat varieties are adjacent in regional wheat trials, the plots of high-stem wheat varieties are subject to marginal positive effects, resulting in a significant increase in economic yield; the plots of short-stem wheat varieties are subject to marginal negative effects, resulting in a decrease in economic yield. This study reveals the mutual influence mechanism of environment and yield with adjacent planting of high-stem and short-stem wheat varieties in regional wheat trials, providing a useful reference and guidance for optimizing the layout of regional wheat trials.

This study examines the marginal effect of Research and Development (R&D) investment and impacts of market liberalization on patenting activities of Japan’s nine incumbent electric power companies. We apply the negative binomial panel data regression model to a data set, comprising of companies from 1999 to 2018 and estimate four models. We find the following significant outcomes. First, retail market liberalization for high voltage consumers proves effective to increase patent applications. Second, R&D investment produces patent applications or a positive marginal effect of R&D on patenting is indicated. These results are consistent with previous findings in a way that deregulation to a certain extent facilitates innovation of firms but it may reverse the effect and decrease inventive activities after a threshold point. In addition, the results show a positive marginal effect of R&D investment on innovations; but the degree of the marginal effect declines with retail market liberalization for high-voltage consumers. This finding implies that innovation efficiency decreases due to the progress of deregulation. This result has critical policy implications; government policies for stimulating inventive activities of electric power companies are necessary and these should ultimately benefit consumers with advanced technology and reasonable prices for energy services.

Methods textbooks in sociology and other social sciences routinely recommend the use of the logit or probit model when an outcome variable is binary, an ordered logit or ordered probit when it is ordinal, and a multinomial logit when it has more than two categories. But these methodological guidelines take little or no account of a body of work that, over the past 30 years, has pointed to problematic aspects of these nonlinear probability models and, particularly, to difficulties in interpreting their parameters. In this review, we draw on that literature to explain the problems, show how they manifest themselves in research, discuss the strengths and weaknesses of alternatives that have been suggested, and point to lines of further analysis.

Multiplicative interaction models are widely used in social science to examine whether the relationship between an outcome and an independent variable changes with a moderating variable. Current empirical practice tends to overlook two important problems. First, these models assume a linear interaction effect that changes at a constant rate with the moderator. Second, estimates of the conditional effects of the independent variable can be misleading if there is a lack of common support of the moderator. Replicating 46 interaction effects from 22 recent publications in five top political science journals, we find that these core assumptions often fail in practice, suggesting that a large portion of findings across all political science subfields based on interaction models are fragile and model dependent. We propose a checklist of simple diagnostics to assess the validity of these assumptions and offer flexible estimation strategies that allow for nonlinear interaction effects and safeguard against excessive extrapolation. These statistical routines are available in both R and STATA.

Many research questions involve comparing predictions or effects across multiple models. For example, it may be of interest whether an independent variable's effect changes after adding variables to a model. Or, it could be important to compare a variable's effect on different outcomes or across different types of models. When doing this, marginal effects are a useful method for quantifying effects because they are in the natural metric of the dependent variable and they avoid identification problems when comparing regression coefficients across logit and probit models. Despite advances that make it possible to compute marginal effects for almost any model, there is no general method for comparing these effects across models. In this article, the authors provide a general framework for comparing predictions and marginal effects across models using seemingly unrelated estimation to combine estimates from multiple models, which allows tests of the equality of predictions and effects across models. The authors illustrate their method to compare nested models, to compare effects on different dependent or independent variables, to compare results from different samples or groups within one sample, and to assess results from different types of models.

Food and nutrition insecurity remains a pressing challenge in many emerging economies. This study examines the heterogeneous impacts of greenhouse farming (GHF) on household food insecurity, dietary diversity, and food consumption in Ghana. Using survey data from 400 vegetable‐producing households and applying marginal and policy‐relevant treatment effect (MTE and PRTE) models, the analysis reveals significant heterogeneity in gains from GHF, shaped by both observable and unobservable household characteristics. Overall, GHF adoption is associated with increased dietary diversity and food consumption, as well as reduced food insecurity. The PRTE estimates indicate that improving farmers' access to produce markets could raise household dietary diversity and food consumption by 42% and 41%, respectively, while lowering food insecurity by 25%. By quantifying both the heterogeneous impacts of GHF and the role of market access, this study provides new evidence on how climate‐smart agricultural technologies can enhance household nutrition and food security in sub‐Saharan Africa.

The spatial morphology of waterfront green spaces helps generate cooling effects to mitigate the urban heat island effect (UHI) in metropolis cities. To explore the contribution and influence of multi-dimensional spatial indices on the mitigation of UHIs, the green space of the riparian buffer along 18 river channels in Shanghai was considered as a case study. The spatial distribution data of the land surface temperature (LST) in the study area were obtained by using remote sensing images. By selecting the related spatial structure morphological factors of the waterfront green space as the quantitative description index, the growth regression tree model (BRT) was adapted to analyze the contribution of various indexes of the waterfront green space on the distribution of the LST and the marginal effect of blue–green synergistic cooling. In addition, mathematical statistical analysis and spatial analysis methods were used to study the influence of the morphological group (MG) types of riparian green spaces with different morphological characteristics on the LST. The results showed that in terms of the spatial structure variables between blue and green spaces, the contribution of river widths larger than 30 m was more notable in decreasing the LST. In the case of a larger river width, the marginal effect of synergistic cooling could be observed in farther regions. The green space that had the highest connectivity degree and was located in the leeward direction of the river exhibited the lowest LST. In terms of the spatial morphology, the fractional cover values of the vegetation (Fv) and area (A) of the green space were the main factors affecting the cooling effect of the green space. For all MG types, a large green patch that had a high green coverage and connectivity degree, as well as was distributed in the leeward direction of the river, corresponded to the lowest LST. The research presented herein can provide methods and development suggestions for optimizing spatial thermal comfort in climate adaptive cities.

The partial (ceteris paribus) effects of interest in nonlinear and interactive linear models are heterogeneous as they can vary dramatically with the underlying observed or unobserved covariates. Despite the apparent importance of heterogeneity, a common practice in modern empirical work is to largely ignore it by reporting average partial effects (or, at best, average effects for some groups). While average effects provide very convenient scalar summaries of typical effects, by definition they fail to reflect the entire variety of the heterogeneous effects. In order to discover these effects much more fully, we propose to estimate and report sorted effects—a collection of estimated partial effects sorted in increasing order and indexed by percentiles. By construction, the sorted effect curves completely represent and help visualize the range of the heterogeneous effects in one plot. They are as convenient and easy to report in practice as the conventional average partial effects. They also serve as a basis for classification analysis, where we divide the observational units into most or least affected groups and summarize their characteristics. We provide a quantification of uncertainty (standard errors and confidence bands) for the estimated sorted effects and related classification analysis, and provide confidence sets for the most and least affected groups. The derived statistical results rely on establishing key, new mathematical results on Hadamard differentiability of a multivariate sorting operator and a related classification operator, which are of independent interest. We apply the sorted effects method and classification analysis to demonstrate several striking patterns in the gender wage gap. We find that this gap is particularly strong for married women, ranging from -60% to 0% between the 2% and 98% percentiles, as a function of observed and unobserved characteristics; while the gap for never married women ranges from -40% to +20%. The most adversely affected women tend to be married, do not have college degrees, work in sales, and have high levels of potential experience.