Why Tables?

It is not difficult to discern why researchers choose to present empirical results using tables. Compared to graphs, tables are much easier to produce. In fact, it is often possible to convert statistical output automatically into a typeset-quality table using a single command.¹⁰

In addition, tables are standard in teaching, presentation, and publishing, thereby providing incentives for scholars to continue producing them. Finally, since tables communicate precise numbers, they are valuable for aiding replication studies (a point we return to in our conclusion).

At the same time, it is easy to understand why researchers are reluctant to use graphs. For one, it simply takes more work to produce graphs. With current software, greater knowledge of the nuances of the statistical/graphical packages is needed to produce effective graphs.¹¹

More importantly, creating informative statistical graphs involves repeated iterations, trial-and-error and much thought about both the deeper issue of what message the researcher is trying to convey and the practical issue of producing a graph that effectively communicates that message. This process can be quite time-consuming; simply put, it takes much greater effort to produce a quality graph than a table.¹²

Another reason why researchers hesitate to use graphs may be their belief that it is simply not feasible to present certain information graphically. Relatedly, some may believe that graphs take up much more space than tables. Both of these concerns are likely to be paramount particularly with respect to regression tables, which can include multiple models that may involve various combinations of variables, observations, and estimation techniques. Researchers may believe it impossible to present results from regressions graphically.

Why Graphs?

We argue, however, that the costs of producing graphs are outweighed by the benefits, and many of the concerns regarding their production are either overstated or misguided altogether. While producing graphs does require greater effort, the very process of graph creation is one of the main benefits of using graphs instead of tables in that it provides incentives for the researcher to present the results more directly and cleanly. Like Gelman et al., we struggled with several versions of each graph presented in this paper before settling on the versions that appear.¹³

While at times frustrating, the iterative process forced us to carefully consider our communications goals and the means of accomplishing them with each graph. Such iteration, of course, is not needed with tables. Thus, the very strength of tables (their ease of production) can also be seen as a weakness.

In addition, concerns about the infeasibility of graphs when presenting certain numeric summaries and about the size of graphs relative to tables are unwarranted. As we illustrate, it is not only possible to present regression results—including multiple specifications—in graphical form, but it is desirable as well. In addition, most of our graphs take up no more room than the tables they replace, including regression tables. And for those that do, we believe the benefit of graphical presentation outweighs the cost of greater size.

Once performed, the extra work put into producing graphs can reap large benefits in communicating empirical results. Extensive experimental research has shown that when the presentation goal is comparison (as opposed to communicating exact values, for which tables are superior), good statistical graphs consistently outperform tables.¹⁴

Our goal in this paper is not to add to the voluminous literature systematically investigating the virtues of graphs versus tables. Rather, our approach is practical; we take a sample of representative tables from political science journals, present them graphically, and then qualitatively compare the two. We believe that these examples illustrate that graphs are simply better devices than tables for making comparisons, which is almost always the goal when presenting data and empirical results. And for those who are convinced, we have created a web site (www.tables2graphs.com) that contains complete replication code for producing our graphs, which can help researchers turn their tables into graphs.

Using a Mosaic Plot to Present Cross Tabulations

We begin with a table presented in Iversen and Soskice, whose study of redistribution in advanced democracies includes nine tables, seven of which present numerical information.¹⁵

Of these, five present descriptive statistics and two present regression results. It is commendable that the authors chose to present in detail much of the data used; notably, however, they chose not to use a single graph in their article.

Their first table (reproduced in our table 1) presents a cross tabulation of electoral systems and government partisanship for a sample of advanced democracies. The key comparison here is whether majoritarian electoral systems are more likely to feature right governments and proportional representation systems are more likely to feature left governments, a comparison presented in numeric form in the last column. The raw numbers that go into this comparison are presented in the main columns. Although the information in this 2-by-2 table is relatively easy to digest, we think that the same information can be more clearly and succinctly presented by using a mosaic plot, a type of graph that is specifically designed to represent contingency tables graphically.¹⁶

The top plot depicts the relationship between electoral system and partisanship of government, while the bottom plot depicts how often countries featured center-left governments at least 50 percent of the time, broken down by their electoral systems. A key feature of the mosaic plot (in contrast to a standard bar plot, for example) is that the area of each rectangle is proportional to the number of observations that fall within their respective contingencies. Thus, the larger width of the rectangles under proportional systems indicates that the majority of countries in the sample have such systems.

The first thing to note about the graph is that the main comparison the authors are attempting to make immediately stands out, as the graph shows that proportional systems are significantly more likely to produce left governments, and that nearly every country with a proportional system featured center-left governments more than 50 percent of the time from 1945–1998 (defined as an “overweight” in their paper), while no countries with a majoritarian system featured center-left governments more than 50 percent of the time. We add the raw counts from each category inside the rectangles; the use of a mosaic plot allows us to combine the best feature of a table—its ability to convey exact numbers—with the comparative virtues of a graph. While actual values are frequently not of interest (and hence do not need to be displayed in a graph), the inclusion of counts here alerts the reader that small samples are involved in the calculation of countries with an overweight of left countries without either adding unnecessary clutter to the graph or increasing its size. Finally, the titles above each plot make it clear what is being plotted, while one has to read the actual text in the original article to understand the table.

This example illustrates that even simple and easy-to-read tables can be improved through graphical presentation. As the complexity of a table grows, the gains from graphical communication will only increase. Indeed, mosaic plots can easily be extended to display multidimensional contingency relationships.¹⁷

Using a Dot Plot to Present Means and Standard Deviations

Another common type of table is one presenting descriptive statistics about central tendencies (e.g., means) and variation (e.g., standard deviations). As we discuss below, presenting only means and standard deviations (along with minimums and maximums) may not be the best choice, depending on the nature of the variables involved. However, even when they are sufficient, it can be difficult to make comparisons across variables and to inspect the distribution of individual variables when data is presented tabularly. Graphs, on the other hand, accomplish both goals.

As an example, we turn to Table 1, panel A, from McClurg (reproduced in our table 2), which presents summary statistics from his study of the relationship between social networks and political participation.¹⁸

We transform the table into a modified dot plot, which is very well suited for presenting descriptive information.¹⁹ We use a single plot, taking advantage of the fact that the scales of all variables are similar. The dots depict the means of each variable, the solid line extends from the mean minus one to the mean plus one standard deviation, and the dashed line extends from the minimum to the maximum of each variable. Rather than ordering the variables randomly or alphabetically, we order them by their mean values, in descending order, which eases comparison of variables.²⁰ Because the number of respondents covered under each variable is not a feature of the variables themselves, we note them under the labels on the y-axis (if we were interested in comparing sample sizes across variables, we could either include a separate graph or make the area of the dots proportional to sample size). The benefits of the graph are apparent: the dots allow for easy lookup and comparison of the means, while the lines visually depict the distribution of each variable. For example, the graph reveals that political talk is right-skewed, which is not easily concluded from the table.

Using Dot Plots and Violin Plots to Present Distributions of Variables with Differing Scales

In many instances, it will be worthwhile to go beyond simple descriptions of variables and present more complete summaries of distributions. As Cleveland and McGill note, “Graphing means and sample standard deviations, the most commonly used graphical method for conveying the distributions of groups of measurements, is frequently a poor method. We cannot expect to reduce distributions to two numbers and succeed in capturing the widely varied behavior that data sets in science can have.”²¹

For an example of when it is useful to depict fuller representations of variables, we turn to table 2 of Kaplan et al. (reproduced in our table 3), which displays summary statistics from their analysis of issue convergence and campaign competitiveness.²²

This table presents a challenge for graphical representation that did not arise in the previous example: the scales of the variables differ dramatically. A quick check of the table reveals that the mean, variance, and maxima differ significantly across variables; the mean issue convergence and percent negative ads, for example, are much higher than the other variables. Including all the variables in a single graph would result in severe compression in a majority of the variables, rendering the graph uninformative.

Scale difference is likely to occur in many datasets, and such situations will require the analyst to be creative about the best way to present her data. We suggest two options, the first of which we pursue here. Instead of using a single graph, we decided to group similar variables and present separate graphs for each group, allowing for comparisons within each graph.

There are three main categories of variables in Kaplan et al.'s table 2: binary variables, those measured in millions, and those measured in percents (including issue convergence, which is defined as the percentage of combined attention that Democratic and Republican candidates give to a particular issue). Since competitiveness and issue salience do not fall into any of these categories, we group them with similarly distributed variables.

We begin the construction of our graph, presented in figure 4, by considering what type of display is best suited for each group of variables. Because binary variables can take on only two values, their distribution is fully characterized by their means and sample size. Accordingly, in the top panel, we plot the means of the binary variables, ordering the variables in descending order and placing sample size in the y-axis labels. (We use diamonds to distinguish them from the median values plotted in the two panels below.) This allows for clear comparison of how often each variable is coded as “1.”

We next consider the variables measured in percents and those measured in millions. While on different scales, both sets of variables are continuous. A range of options exist for graphing the distribution of continuous variables, including histograms, kernel density plots, and boxplots.²³

A recent graphical innovation called the “violin plot” combines the virtues of the latter two by overlaying a density trace onto the structure of a box plot.²⁴ The resulting plot presents both central tendencies and detailed information on the distribution of variables, including whether they are skewed and the presence of outliers. For each set of variables, we present a panel including violin plots for each variable. The center of each violin plot gives information similar to that presented in a traditional box-plot: the points depict median values, the white boxes connect the 25th and 75th percentiles and the thin black lines connect the lower adjacent value to the upper adjacent value, which help identify skewness or the presence of outliers or both.²⁵ The shaded area depicts a density trace of each variable, which is plotted symmetrically above and below the horizontal box plot in order to aid visualization. As with a kernel density plot, a hill distribution (such as “State Voting Age Pop.” in figure 4) indicates normality, while a rectangle distribution (such as “Competitiveness”) indicates a more uniform distribution.

The violin plots reveal several characteristics of the data that are not apparent from the table. First, many of the variables exhibit substantial skewness. For instance, while the median of both “Issue Convergence” and “Issue Salience” are 0, their tails extend well to the right. In addition, the presence of observations well beyond the upper adjacent values of “Total Spending/Capita” and “Difference Spending/Capita” indicates the existence of outliers (indeed, there is only one observation of the latter that is greater than four).

These are details that one would be hard-pressed to glean from a table, even one that went beyond reporting only means and standard deviations. Greater understanding of distributions can aid researchers as they move from exploratory data analysis to model fitting. And, as these violin plots illustrate, they can increase the reader's understanding of the data that is presented and analyzed.

Using an Advanced Dot Plot to Present Multiple Comparisons

The second option for graphing descriptive plots of variables with different scales is to alter the scales themselves. We illustrate this option by converting table 2 from Schwindt-Bayer's study of the attitudes and bill initiation behavior of female legislators in Latin America (reproduced in our table 4) into a graph.²⁶

Most of the rows of the table are devoted to displaying the number of bills introduced by lawmakers in four Latin American legislative chambers across two time periods (which vary by chamber). While the text of the paper does not explicitly state what comparisons the reader should draw from the table, there are three main comparisons one could make: across issue areas, across countries, and across time periods. The structure of the table, however, only allows for easy comparison of issue areas—for a single country and a single time period. The use of absolute counts for the number of bills introduced in each issue area instead of percentages hinders cross-column comparisons (i.e. across countries and time periods). Using percentages would improve the presentation, but would still place the burden on the reader to find patterns in the data.

Instead, we turn the table into an advanced dot plot, presented in figure 5. We begin by converting the counts of each bill introduced in each issue area into a proportion, with the total number of bills initiated serving as the denominator. For each chamber and for each issue area, we present the proportions for both the first and second time periods using different symbols: open circles for the first period, “+”s for the second. (Because the “number of legislators who sponsored at least one bill” and the “total number of bills initiated” seem tangential to the table, we include them only as counts under the name of each country or chamber in the panel strips.) One option, of course, would be simply to scale the x-axes on a linear proportion scale, ranging from 0 to 1; this, however, would mask differences among several issue areas for which relatively few bills were introduced.²⁷

Instead, we follow the advice of Cleveland and scale the x-axes on the log 2 scale, and provide for easy lookup by placing tick mark labels at both the top and bottom of the graph.²⁸ Given this scaling, each tick mark (indicated vertically by the solid gray lines) represents a proportion double that of the tick mark to the left. Looking at the Colombia Senate, for example, we can see that the proportion of bills related to health issues declined by half from the 1994–1998 period to the 1998–2002 period. We can also compare easily across issue areas: in Argentina in 1999, for example, it is easy to see that twice as many fiscal affairs bills were introduced as women's issues bills.

The graph thus allows for all three types of comparisons. The use of dual symbols allows for easy comparison both across time periods within a given issue area and across issue areas within a single time period. The graph also facilitates cross-legislature comparison by placing all plots in a single column; by reading vertically down the graph, for example, we can see that proportionally more bills pertaining to women's issues were introduced in the Colombia Senate than the Colombia Chamber from 1998–2002. This plot, from the choice of scaling to the tool we utilized, follows the principles outlined in Cleveland, showcasing the ability of Trellis plots to allow comparisons within and across panels.²⁹

The technique of using several smaller graphs instead of one big graph—coined “small multiples” by Tufte—can greatly enhance the presentation of data and empirical results in a wide range of situations.³⁰

In summary, we believe that these examples demonstrate the benefits of graphing descriptive statistics. Of course, there are many other types of descriptive data summaries, such as correlation matrices and time series, that will require different graphing strategies.³¹

Our purpose is not to exhaust all of these, but to illustrate how descriptive tables can be turned into graphs with great benefit.

On Regression Tables and Confidence Intervals

Regression tables are meant to communicate two essential quantities: point estimates (coefficients) and uncertainty estimates (usually in the form of standard errors, confidence intervals, or null hypothesis tests). In our sample, 74% of the regression tables presented standard errors, usually supplemented by asterisks labeling coefficients that have attained conventional levels of statistical significance. Thus, tables typically seek to draw attention to coefficients that are significant at the p < .01, p < .05 or p < .10 levels.

This tendency likely stems from the discipline's reliance on null hypothesis significance testing in conducting regression analyses. Articles in fields as diverse as wildlife management, psychology, medicine, statistics, forecasting, and political science argue that null hypothesis significance testing and reliance on p-values can lead to serious mistakes in statistical inference.³²

These articles suggest substituting confidence intervals for p-values and significance testing when presenting regression results.

These criticisms notwithstanding, it is likely that political scientists will continue to rely on null hypothesis significance testing. Given this reality, can graphs improve on standard regression tables? We believe the answer is yes, due to the simple fact that graphs of point estimates and confidence intervals can communicate the same information as standard regression tables while adding the virtues of emphasizing effect size and easy comparison of coefficients. A confidence interval effectively presents the same information as a null hypothesis significance test; for example, a 95 percent confidence interval that does not include the null hypothesis (typically zero) is equivalent to a coefficient being statistically significant at p < .05.³³

In addition, “confidence intervals have a great virtue: as the sample size increases the size of the interval decreases, correctly expressing our increased certainty about the parameter of interest.”³⁴

Of course, it is possible to present confidence intervals in regression tables.³⁵

Notably, however, none of the tables in our sample featured confidence intervals. While the discipline's reliance on null hypothesis testing is perhaps the main cause of this tendency, the practicality of incorporating confidence intervals may also play a role: it is simply difficult to present such tables, in particular when multiple specifications or subsamples are compared (as occurred in about 88 percent of the regression tables in our sample). Consider table 4 of Ansolabehere and Konisky, which we replicate in table 5 (we present the graphical version of the table later in this section).³⁶ The authors estimate the effect of voter registration laws on county-level turnout in New York and Ohio, and do robustness checks by presenting six different models, as seen in columns one through six in the table. Note that there are about three dozen coefficients and standard errors pairs to compare.

Would the use of confidence intervals improve interpretation? In table 6, we present the same table, but instead of standard errors and asterisks we use confidence intervals. Although inferences are more direct when estimates are presented in this form, it gets confusing quickly when comparing across models. Even a simple question, such as which confidence intervals overlap, demands careful attention to signs, and comparisons must be done one by one.

These difficulties point to a main advantage of the standard regression table: it is less confusing than the alternative. There are others: it is clear from the table which independent variables are present or missing in each model. In addition, information such as model fit statistics and the number of observations can easily be added to the table. In summary, regression tables are able to display a large wealth of information about the models in a very compact and mostly readable format. They also clearly communicate the results of null hypothesis significance testing. It is no surprise that they are so popular.

Could graphs do a better job? As we illustrate below, graphs can easily display point estimates and confidence intervals, including those from multiple models. In doing so, they can clearly convey the results of null significance hypothesis testing. And once we move beyond the sole consideration of whether a coefficient is significantly different from zero, we believe the benefits of graphs become even more apparent, allowing for the proper highlighting of effect size and comparison of coefficients both within and across models.

In summary, as we move to graphing regression results, and given the aforementioned strengths and weaknesses of tables, we look for a good regression graph to satisfy the following criteria:

1. It should make it easy to evaluate the statistical significance of coefficients;
2. It should also be able to display several regression models in a parallel fashion (as tables currently do);
3. Relatedly, when models differ by which variables are included, it should be clear to the reader which variables are included in which models;
4. It should be able to incorporate model information;
5. Finally, the plot should focus on confidence intervals and not (only) on p-values.

Plotting a Single Regression

We start by plotting a simple regression table. Table 2 from Stevens et al. (reproduced in our table 7) displays results from a single least squares regression.³⁷

Using a survey of elites from six Latin American countries, the authors study the effects of economic perceptions, ideology and demographic variables and a set of country dummies on the survey respondent's “individual authoritarianism,” measured on a seven-point scale. The table condenses a large wealth of information: regression fit summaries, the number of cases, point estimates, standard errors, significance tests for multiple comparisons among the country dummies, and asterisks denoting .01, .05 and .10 two-tailed p-values.

In figure 6 we condense the same information into a simple dot plot, much like those used in the previous section. We take advantage of the similar scaling across the estimates and display the results in a single plot. The dots represent the point estimates, while the horizontal lines depict 95 percent confidence intervals.³⁸

We place a vertical line at zero for convenience, and make the length of the x-axis symmetric around this reference line for easy comparison of the magnitude of positive and negative coefficients. Each independent variable is displayed on the y-axis. Finally, we omit the estimate and standard error of the constant, since it is not substantively meaningful in this instance (the constant is the predicted value for someone, who, among other unlikely if not impossible characteristics, is age zero.) Finally, we use the empty space in the plot region to display R², adjusted R² and the number of observations; this information, however, could just as well be displayed in the caption, if more desirable or if there is not enough room in the plot region.

This figure shows several advantages of using graphs. First, information regarding statistical significance is displayed without any asterisks, bars or superscripted letters. Instead, the length of the error bars visually signal which variables are significant: those that do not cross the reference line, which is zero in this case. Thus, a vertical scan of the graph allows the reader to assess quickly which variables are significantly different from the null hypothesis.

In addition, the visual display of the regression results also focuses the attention of the reader away from statistical significance towards the more relevant and perhaps more interesting information revealed by a regression analysis: the estimated effect size and the degree of uncertainty surrounding it. The vertical placement of coefficients makes it easy to compare their relative magnitudes, while the size of the error bars provide information on how precisely each parameter is estimated.

The use of confidence intervals also provides much more information, displayed intuitively, than a regression table. For instance, when confidence intervals do not overlap, we can conclude that two estimates are statistically significantly different from each other. Thus, if one goal of a regression analysis is to compare two or more parameter estimates, then this simple type of graph is sufficient. However, even if the confidence intervals of two coefficients overlap, it is possible that they are still significantly different.³⁹

One possibility would be to move beyond this basic graph and plot all contrasts of interest (the estimates of the differences in the country coefficients and their corresponding standard errors, for example: b_Argentina−b_Chile;b_Argentina−b_Colombia, etc.).⁴⁰

Plotting Multiple Models in a Single Plot

As noted earlier, in our survey of political science tables we found that more often than not researchers present multiple models. Graphing multiple models presents new challenges, as we must be sure that a reader can distinguish the parameter estimates and confidence intervals from each model and that the differences between models in terms of which variables are included are visually apparent.

We begin with the case of two regression models. Table 1 of Pekkanen et al. (replicated in our table 8) displays two logistic regression models that examine the allocation of posts in the LDP party in Japan.⁴¹

In the first model, PR Only and Costa Rican in PR are included and Vote share margin is excluded, while in the second model the reverse is true. We present the two models by plotting parallel lines for each of them grouped by coefficients, as can be seen in figure 7. We differentiate the models by plotting different symbols for the point estimates: filled (black) circles for the first model and empty (white) circles for the second. The similar scaling of the coefficients again allows us to graph all the estimates in a single plot. Because most of the coefficients fall below zero, to save space we do not make the x-axis symmetric around zero. And because the plot region features little empty space, in this case we would present model information in the caption. Unlike in the previous example, we include the constant in the graph, since it is substantively meaningful in these models (the inverse logit of the constant in each model gives the predicted probability that a third-term Member of Parliament who is coded as 0 on all the other predictors in the model—theoretically possible given that they are all categorical variables—holds a senior political position.

By plotting the two models, we can now easily compare both the coefficients within each model and across the two models. The fact that only single estimates are plotted for PR Only, Costa Rican in PR and Vote share margin signals that these predictors appear in only one model. Rather than having to compare individual parameter estimates (and the presence or absence of asterisks) across models in a regression table, a vertical scan of the graph shows that the estimates are mostly robust across models, as the parameter estimates and their respective confidence intervals vary little. Finally, plotting the coefficients for the term indicators shows that LDP members in their first term are much less likely to receive leadership posts, all things equal, than members in their third terms, an intuitive result that does not necessarily stand out in the crowded regression table.

Using Multiple Plots to Present Multiple Models

As the number of models increase, we suggest a different strategy. Instead of presenting all the models and predictors in a single plot, we use “small multiple” plots to present results separately for each predictor. The main objective when presenting results from multiple models is to explore how inferences change as we change the model specification or use different subsamples, or both. Using multiple dot plots with error bars allows the researcher to communicate the information in a multiple regression table with much greater clarity and also make it much easier to compare parameter estimates across models. This strategy also overcomes any problems that might arise from having predictors with greatly varying scales, since the coefficients from each independent variable are presented in a separate plot.

As an example, we return to table 4 of Ansolabehere and Konisky (see table 5 above), turning it into the graph presented in figure 8.⁴²

Rather than placing all the predictors in a single plot, which would make it difficult to compare individual estimates, we created a separate panel for each of the six predictors, with the panels presented in a single column. We also shift strategy and display the parameter estimates on the y-axes, along with 95 percent confidence intervals. This maximizes efficiency since we can stack more plots into a single column than into a single row, given the portrait orientation of journals. The x-axis depicts which model is being displayed. To facilitate comparison across predictors, we center the y-axis at zero, which is the null hypothesis for each of the predictors.

The regression table presents six models, which vary with respect to sample (full sample vs. excluding partisans registration counties) and predictors (with/without state year dummies and with/without law change). On the x-axis we group each type of model: “full sample,” “excluding counties with partial registration” and “full sample with state year dummies.” Within each of these types, two different regressions are presented: including the dummy variable law change and not including it. Thus, for each type, we plot two point estimates and intervals—we differentiate the two models by using solid circles for the models in which law change is included and empty circles for the models in which it is not. We again choose not to graph the estimates for the constants because they are not substantively meaningful.

This graphing strategy allows us to easily compare point estimates and confidence intervals across models. Although in all the specified models the percent of county with registration predictor is statistically significant at the 95 percent level, it is clear from the graph that estimates from the full sample with state/year dummies models are significantly different from the other four models. In addition, by putting zero at the center of the graph, it becomes obvious which estimates have opposite signs depending on the specification (log population and log median family income). By contrast, it is much more difficult to spot these changes in signs in the original table. Thus, by using a graph it is easy to visually assess the robustness of each predictor—both in terms of its magnitude and confidence interval—simply by scanning across each panel. In summary, the graph appropriately highlights the instability in the estimates depending on the choice of model.