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4: How to Report Statistics

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    Introduction

    While you are thinking about exploring data sets and descriptive statistics, please review our overview of data analysis (Chapter 2.4 and 2.5). While the scientific hypotheses come first, how experiments are designed should allow for straight-forward analysis: in other words, statistics can’t rescue poorly designed experiments, nor can it reveal new insight after the fact.

    Once the experiments are completed, all projects will go through a similar process.

    • Description: Describe and summarize the results
    • Check assumptions
    • Inference: conduct tests of hypotheses
    • Develop and evaluate statistical models

    Clearly this is a simplification, but there’s an expectation your readers will have about a project. Basic questions like how many subjects got better on the treatment? Is there an association between Body Mass Index (BMI) and the primary outcome? Did male and female subjects differ for response to the treatment? Undoubtedly these and related questions form the essence of the inferences, but providing graphs to show patterns may be as important to a reader as any p-value — a number which describes how likely it is that your data would have occurred by chance — e.g., from an Analysis of variance.

    Each project is unique, but what elements must be included in a results section?

    Data visualization

    We describe data in three ways: graphs, tables, and in sentences. In this page we present the basics of when to choose a graph over presenting data in a table or as a series of sentences (i.e., text). In the rest of this chapter we introduce the various graphics we will encounter in the course. Chapter 4 covers eight different graphics, but is by no means an exhaustive list of kinds of graphs. Phylogenetic network graphs are presented in Chapter 20.11. Although an important element of presentation in journal articles, we don’t discuss figure legends or table titles; guidelines are typically available by the journal of choice (e.g., PLOS ONE journals guidelines).

    A quick note about terminology. Data visualization encompasses charts, graphs and plots. Of the three terms, chart is the more generic. Graphs are used to display a function or mapping between two variables; plots are kinds of graphs for a finite set of points. There is a difference among the terms, but I confess, I won’t be consistent. Instead, I will refer to each type of data visualization by its descriptive name: bar chart, pie chart, scatter plot, etc. Note that technically, a scatter plot can refer to a graph, e.g., a line drawn to reflect a linear association between the two variables, whereas bar charts and pie charts would not be a graph because no function is implied.

    Why display data?

    Do we just to show a graph to break the monotony of page after page of text, or do we attempt to do more with graphs? After all, isn’t “a picture worth a thousand words?” In many cases, yes! Graphics allow us to see patterns. Visualization is a key part of exploratory data analysis, or data mining in the parlance of big data. In genomics, heat maps

    Graphics are complicated and expensive to do well. Text is much cheaper to publish, even in digital form. But the ability to visualize concepts, that is, to connect ideas to data through our eyes (see Wikipedia), seems to be more the cognitive goal of graphics. Lofty purpose, desirable goal. Yes, it is true that graphics can communicate concepts to the reader, but with some caution. Images distort, and default options in graphics programs are seldom acceptable for conveying messages without bias (Glazer 2011).

    Here’s some tips from a book on graphical display (Tufte 1983; see also Camões 2016).

    Your goal is to communicate complex ideas with clarity, precision, and efficiency. Graphical displays should:

    • show the data
    • avoid distorting the data
    • present numbers in a small space
    • help the viewer’s eye to compare different pieces of data
    • serve a clear purpose (description, exploration, tabulation, decoration)
    • be closely integrated with statistical and verbal descriptions of a data set.

    We accomplish these tasks by following general principles involving scale and a commitment to avoiding bias in our presentation.

    Importantly, graphs can show patterns not immediately evident in tables of numbers. See Table \(\PageIndex{1}\) for an example of a dataset, “Anscombe’s quartet,” (Anscombe 1973), where a picture is clearly helpful.

    Table \(\PageIndex{1}\). Anscombe's data (Anscombe 1973).
    X Y1 Y2 Y3 Y4
    10 8.04 9.14 7.46 6.58
    8 6.95 8.14 6.77 5.76
    13 7.58 8.74 12.74 7.71
    9 8.81 8.77 7.11 8.84
    11 8.33 9.26 7.81 8.47
    14 9.96 8.10 8.84 7.04
    6 7.24 6.13 6.08 5.25
    4 4.26 3.10 5.39 12.50
    12 10.84 9.13 8.15 5.56
    7 4.82 7.26 6.42 7.91
    5 5.68 4.74 5.73 6.89
    Mean (±SD) 7.50 (2.032) 7.50 (2.032) 7.50 (2.032) 7.50 (2.032)
    Note that the data set does not include the column summary statistics shown in the last row of the table.

    The Anscombe dataset is also available in R package stats, or you can copy/paste from Table \(\PageIndex{1}\) into a spreadsheet or text file, then load the data file into R (e.g., Rcmdr → Load data set). Note that the data set does not include the column summary statistics shown in the last row of the table.

    Before proceeding, look again at the table — See any patterns in the table?

    Maybe.… Need to be careful as we humans are really good at perceiving patterns, even when no pattern exists.

    Now, look just at the last row in the table, the row containing the descriptive statistics (the means and standard deviations). Any patterns?

    The means and standard deviations are the same, so nothing really jumps out at you — does that mean that there are no differences among the columns, then?

    But let’s see what the scatter plots look like before we conclude that the columns of Y ’s are the same (Fig. \(\PageIndex{1}\)). I’ll also introduce the R package clipr, which is useful for working with your computer’s clipboard.

    Definition: Term

    To show clipboard history, on Windows 10/11 press Windows logo key plus V; on macOS, open Finder and select Edit → Show Clipboard.

    Four scatter plots, each plotting Anscombe's X data against one of Anscombe's columns of Y data.
    Figure \(\PageIndex{1}\): Scatter plot graphs of Anscombe’s quartet (Table \(\PageIndex{1}\)).
    #R code for Figure 1.
    
    require(clipr)
    #Copy from the Table and paste into spreadsheet (exclude last row). Highlight and copy data in spreadsheet
    myTemp <- read_clip_tbl(read_clip(), header=TRUE, sep = "\t")
    #Check that the data have been loaded correctly
    head(myTemp)
    #attach the data frame, so don't have to refer to variables as myTemp\$variable name
    attach(myTemp)
    #set the plot area for 4 graphs in 2X2 frame
    par(mfrow=c(2,2))
    plot(X, Y1, pch=19, col="red", cex=1.2)
    plot(X, Y2, pch=19, col="orange",cex=1.2)
    plot(X, Y3, pch=19, col="yellow",cex=1.2)
    plot(X, Y4, pch=19, col="blue",cex=1.2)

    And now we can see that the Y ‘s have different stories to tell. While the summary (descriptive) statistics are the same, the patterns of the association between Y values and the X variable are qualitatively different: Y1 is linear, but diffuse; Y2 is nonlinearly associated with X; Y3, like Y1, is linearly related to X, but one data point seems to be an outlier; and for Y4 we see a diffuse nonlinear trend and an outlier.

    So, that’s the big picture here. In working with data, you must look at both ways to “see” data — you need to make graphs and you also need to calculate basic descriptive statistics.

    And as to the reporting of these results, sometimes tables are best (i.e., so others can try different statistical tests), but patterns can be quickly displayed with carefully designed graphs. Clearly, in this case, the graphs were very helpful to reveal trends in the data.

    When to report numbers in a sentence? In a table? In a graph?

    The choice depends on the message. Usually you want to make a comparison (or series of comparisons). If you are reporting one or two numbers in a comparison, a sentence is fine. “The two feral goat populations had similar mean numbers (120 vs. 125) of kids each breeding period.” If you have only a few comparisons to make, the text table is useful:

    Table \(\PageIndex{2}\). Data from Kipahoehoe Natural Area Reserve, SW slope of Mauna Loa.
    Location Number of kids
    Outside fence:
    kīpuka 51
    Other 120
    Inside fence:
    kīpuka 3
    Other 15

    To conclude, tables are the best way to show exact numbers and tables are preferred over graphs when many comparisons need to be made. (Note: this was a real data set, but I’ve misplaced the citation!)

    Note:

    From Wikipedia, a kīpuka is a land area surrounded by recent lava flows.

    Couldn’t I use a pie chart for this?

    Yes, but I will try to persuade you not to do so. Pie charts are used to show part-whole relationships. If there are just a few groups, and if we don’t care about precise comparisons, pie charts may be effective. Sometimes, people use pie charts for very small data sets (comparing two populations, or three categories, for example). The problem with pie charts is that they require interpretation of the angles that define the wedges, so we can’t be very precise about that. Bar charts (Chapter 4.1) are much better than pie charts, however.

    To illustrate the problem, here’s a couple of pie charts from Microsoft Excel (a similar chart can be made with LibreOffice Calc) for our goat data set; compare this graph to the table and to the bar chart below (Fig. \(\PageIndex{2}\)).

    Pie chart of Table 2 data, showing that the Outside fence, Other category occupies 63.5% of the total, the Inside fence, Other categories occupies 7.9% of the total, and the remaining 28.6% is made up of 27.0% Outside fence, Kipuka and 1.6% Inside fence, Kipuka.
    Figure \(\PageIndex{2}\): Excel pie chart of Table 2 data set.

    A bar chart of the same data (Fig. \(\PageIndex{3}\)):

    A bar chart showing the data from Table 2. Two categories are shown on the x-axis, "Kipuka" and "Other," and the y-axis shows percentage of the total number of goats. Each category has one bar showing the percentage of goats found inside the fence in that location, and one bar showing the percentage found outside the fence in that location.
    Figure \(\PageIndex{3}\): Bar chart of Table \(\PageIndex{2}\) data set.

    The bar chart (Fig. \(\PageIndex{3}\)) more effectively gets the message across; more goat kids were found outside the fenced area then inside the fenced in areas. We can also see that more goat kids were found in the “other” areas compared to the kipuka. The pie chart (Fig. \(\PageIndex{2}\)), in my opinion, fails to communicate these simple comparisons, which are conclusions about patterns in the data that clearly would be the take-home message from this project. Aesthetically the bar chart could be improved — a mosaic plot would work well to show the associations in the project results (See Chapter 4.4: Mosaic plots).

    But we are not done with this argument on whether to use graphics or text to report results. Neither the bar chart (mosaic plot) or the pie chart really work. The reader has to interpret the graphics by extrapolating to the axes to get the numbers. While it may be boring — 1.5 million hits Google search “data tables” boring — tables can be used for comparisons and make the patterns more clear and informative to the reader. Here’s a different version of the table to emphasize the influence of fencing on the goat population.

    Table \(\PageIndex{3}\). Revised Table \(\PageIndex{2}\) to emphasize comparisons between inside- and outside-the-fence-line feral goat populations on Mauna Loa.
    Location Kipuka Other
    Outside fence 51 120
    Inside fence 3 15

    Table \(\PageIndex{3}\) would be my choice — over a sentence and over a graph. At a glance I can see that more goat kids were found outside of the fenced area, regardless of whether it was in a kipuka or some other area on the mountain side. Table \(\PageIndex{3}\) is an improvement over Table \(\PageIndex{2}\) because it presents the comparisons in a 2 X 2 format — especially useful when we have a conditional set.

    For example, it’s useful to show the breakdown of voting results in tables (numbers of votes for different candidates by voter’s party affiliation, home district, sex, economic status, etc.). Interested readers can then scan through the table to identify the comparison they are most interested in. But often, a graph is the best choice to display information. One final point: by judiciously combining words, numbers, and images, you should be able to convey even the most complex information in a clear manner! We will not spend a lot of time on these issues, but you will want to pay some attention to these points as you work on your own projects.

    Some final comments about how to present data

    What your graph looks like is up to you; lots of people have advice (e.g., Klass 2012). But we all know poor graphs when we see them in talks or in papers; we know them when we struggle to make sense of the take-home message. We know them when we feel like we’re missing the take-home message.

    Here’s my basic take on communicating information with graphics.

    • Minimize white space (for example, the scatter plots above could be improved simply by increasing the point size of the data points)
    • Avoid bar charts for comparisons if you are trying to compare more than about three or four things.
    • A graphic in a science report that is worth “a thousand words” probably is too complicated, too much information, and, very likely, whatever message you are trying to convey is better off in the text.

    • 4.1: Bar (column) charts
      Different forms of bar graphs, the situations in which they should be used, and how to create them in R.
    • 4.2: Histograms
      Types of histograms and their use cases. Kernel density estimation. How to create histograms in R and select the appropriate number of bins.
    • 4.3: Box plots
      Box plots and why they should be the default graph for exploring ratio scale data. How to create box plots in R.
    • 4.4: Mosaic plots
      Mosaic plots and their use in displaying associations among categorical variables. Options for creating mosaic plots in R.
    • 4.5: Scatter plots
      Scatter plots as a way to display associations between two quantitative, ratio-scaled variables. Design criteria of scatter plots and how to create them using R.
    • 4.6: Adding a second Y axis
      Adding a second Y axis to scatter plots, to show the association of a single X variable with both Y variables. How to create such plots using R.
    • 4.7: Q-Q plot
      The Q-Q (quartile-quartile) plot as a way to compare probability distributions. How to create this graph in R Commander.
    • 4.8: Ternary plots
      Ternary plots as a method of displaying 3 ratio variables that in total sum up to 1. Creating ternary plots in R, with examples of use cases.
    • 4.9: Heat maps
      Heat maps as a way of presenting data from a matrix. Discussion of use cases, color choices, and how to create heat maps using R.
    • 4.10: Graph software
      Discussion of software options besides R for generating graphics. The same plot type and data set as graphed in several different softwares.
    • 4.11: Chapter 4 References


    This page titled 4: How to Report Statistics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Michael R Dohm via source content that was edited to the style and standards of the LibreTexts platform.

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