In Chapter 2 we looked at ways of displaying data. Having analysed the data it is important to consider how to present the results to put across our message clearly and give the reader enough information to make their own conclusions.

We start by looking at ways of displaying summary statistics (from chapters 3 and 4). These can be helpful to describe the data and the people within your sample in one of the early sections of the results. Summary statistics can also be used for the outcome of a study.

Second, we show how inferential statistics such as standard errors, confidence intervals and p-values can also be presented to enhance interpretation of the results. They can often be presented in graphs alongside summary statistics or even alongside graphs of the raw data (e.g. dot plots).

- Presenting summary statistics: numeric variable
As discussed in chapter 3, summary statistics of a numeric variable include some measure of the centre of the distribution and also measures of the spread. If the data is normally distributed, the mean and standard deviation are appropriate summary statistics and if not, the median, IQR and other percentiles of the data should be used.

Summary statistics are reported for many reasons; (1) to describe the baseline demographics of your sample, (2) to informally check if there are any differences in the demographics between the intervention and control groups you are comparing (i.e. to check for potential confounding factors), (3) or to summarise the outcome variable of a study.

Here are some ways to present summary statistics of a numeric variable.

### Tables

There is often insufficient space to present graphs for all variables in the dataset, so tables are sometimes sufficient. The data should be explored first to ensure the summary statistics you report give a good representation of the data, capture whether there is any skewness and whether there are any outlying points. If there are outlying points, reporting the range is also recommended.

Here are two example tables of summary statistics, similar to what we have seen in other chapters of these notes. The first shows only the summary statistics 'mean ± SD' and the second shows a more diverse range.

Ref:

**Spiller, O. Brad, and B. Paul Morgan. "Antibody-Independent Activation of the Classical Complement Pathway by Cytomegalovirus-Infected Fibroblasts." The Journal of Infectious Diseases, vol. 178, no. 6, 1998, pp. 1597-1603.**Ref:

**Lobdell, Danelle T., et al. "Feasibility of Assessing Public Health Impacts of Air Pollution Reduction Programs on a Local Scale: New Haven Case Study." Environmental Health Perspectives, vol. 119, no. 4, 2011, pp. 487-493.**### Bar charts

Bar charts are not just for categoric data, they can also be used to show summary statistics of numeric variables such as the mean. However, these are bad displays. They lose the data, have no summary of spread and they assume that the mean is a good summary for each subgroup which it may well not be.

Here is an example looking at the quality of life scores of women with coronary heart disease at baseline and after 6 months.

Ref:

**Christian, Allison H., et al. "Predictors of Quality of Life among Women with Coronary Heart Disease." Quality of Life Research, vol. 16, no. 3, 2007, pp. 363-373.**Marginally better as has some measure of spread. However, still assumes normality and doesn't show outliers. The example below shows a bar chart including 'error bars' of the standard deviation of the measurements. It's important to be clear what these error bars represent in the key or legend, so the reader does not confuse them with confidence intervals.

Ref:

**Spiller, O. Brad, and B. Paul Morgan. "Antibody-Independent Activation of the Classical Complement Pathway by Cytomegalovirus-Infected Fibroblasts." The Journal of Infectious Diseases, vol. 178, no. 6, 1998, pp. 1597-1603.**### Box plots

Box plots can be used to display percentiles of the data, including median, IQR and identify any outliers.

Ref:

**Lagerros, Ylva Trolle, et al.****"Estimating Physical Activity Using a Cell Phone Questionnaire Sent by Means of Short Message Service (SMS): a Randomized Population-Based Study." European Journal of Epidemiology, vol. 27, no. 7, 2012, pp. 561-566.**The example on the above comes from a study looking at the physical activity and the differences between responses depending on whether they were asked on paper, via the web or text message. The means are also given, which is not typically reported in a box plot.

The plots show the data is positively skewed and there is little difference between the three groups.

Ref:

**Millqvist, Eva, et al. "Changes in Levels of Nerve Growth Factor in Nasal Secretions after Capsaicin Inhalation in Patients with Airway Symptoms from Scents and Chemicals." Environmental Health Perspectives, vol. 113, no. 7, 2005, pp. 849-852.**In contrast, here is a box plot of the nerve growth factor (NGF) before and after inhalation provocation. The sample sizes of the intervention and control groups are just 13 and 14 respectively. Box plots are not so useful for such low sample sizes. In this example, they are reducing the 13/14 data points into 5 summary statistics - that is not much of a reduction! The authors may have been better presenting the data as dot plots so that they don't lose information.

Also notice in the two examples, the whiskers (the ends of the vertical lines) are defined differently. In the second example, the whiskers are the 10th to 90th percentiles and in the first example they were the 5th to 95th percentiles. Other examples of box plots would show yet more definitions for these whiskers. It is important to make this clear in the key or legend.

### Dot plots

We presented dot plots in chapter 2; these are useful for presenting the raw data without losing any information. Summary statistics can also be superimposed on these plots to help informally visualize the differences between groups. The next example includes the mean superimposed over dot plots. Given the skew in the data, perhaps the mean was not an appropriate summary measure to use - the median should have been used instead.

Ref:

**Kim, Sung-Han, et al. "Prediction of Residual Immunity to Smallpox, by Means of an Intradermal Skin Test with Inactivated Vaccinia Virus." The Journal of Infectious Diseases, vol. 194, no. 3, 2006, pp. 377-384.**### Bar charts vs box plots vs dot plots

The graph you chose has an impact on the reader's interpretation. A study was conducted on university students to assess whether bar charts, box plots and dot plots are interpreted differently even if they come from the same data. Students were presented with data of the battery lives of two different brands and asked to choose which they prefer if any.

Ref:

**Baglin, J and Grant, S 2016, 'Exploring the impact of visual representations of variation on informal statistical inference', in H. MacGillivray, M. A. Martin and B. Phillips (ed.) Proceedings of the 9th Australian Conference on Teaching Statistics, Canberra, Australia, 8-9 December 2016, pp. 4-9.**The graphs that the students were presented with are shown below.

The top row of graphs show there is no real difference between the two battery brands and the bottom ones that battery brand B was significantly better. The bar charts on the left show the mean battery life and the error bars show mean ± SD.

When there was no real difference between the brands (graphs on the top row), students correctly identified this most in the bar chart. When there was a difference, student correctly identified this most in box plot and least in the bar chart.

A bar chart is not recommended to present means, given the results of the study above. Furthermore, bar charts are normally associated with frequencies, proportions and percentages so this can confuse the reader. Just like in chapter 2, graphs should present the results in the simplest way possible.

Graphs showing summary statistics are a great way to impart information, but they should not be used to make statistical inference - we have standard errors, confidence intervals and p-values for this. These inferential statistics should be presented and can also be displayed on graphs.

- Presenting summary statistics: categoric variable
Recall from chapter 2 and 3 that frequencies, percentages and proportions can be presented to summarise a categorical variable. These can be presented using a frequency distribution, bar chart or pie chart (as described in chapter 2). These summary statistics can be used to describe the demographics (i.e. baseline characteristics) of the study sample.

An example of a baseline demographics table is given below. The table comes from a study aiming to look at social and cultural differences in Asians living in two areas within the USA. Therefore, the demographic table below can help to investigate potential confounding factors before the main results section. It is important to include percentages in tables such as this because looking at frequencies between groups can be misleading if the group sizes are imbalanced. There are no large imbalances between the two groups, but there is a slightly higher percentage of Indians and Bangladeshis in the San Francisco Bay Area.

Ref:

**Mukherjea, Arnab, et al. "Social and Cultural Influences on Tobacco-Related Health Disparities among South Asians in the USA." Tobacco Control, vol. 21, no. 4, 2012, pp. 422-428.**Bar charts and pie charts may also be used to present data such as this if there is space in the journal article. However, demographic statistics are of lower importance than the main results, so often a table will suffice.

- Presenting inferential statistics
After baseline demographics have been presented appropriately, the main results section will usually include inferential statistics such as standard errors, confidence intervals to express the level of precision around any estimates and p-values to test any statistical hypotheses.

The way in which the main results are presented depends on the type of data being analysed, how much space there is in the journal, and how complex the results are.

If the results are very simple, a summary statistic can be presented in the text alongside a confidence interval and p-value. If the results are more complex, results may be presented in the form of a table or graph. Here are some examples from different journals.

The first shows a dot plot superimposed with the mean and 95% confidence intervals (again, another example of where means and confidence intervals are presented on skewed data, it happens frequently in the literature!).

Ref:

**Yang, H., Chen, H., Liu, Z., Ma, H., Qin, L., Jin, R., ... & Liu, J. (2013). A novel B-cell epitope identified within Mycobacterium tuberculosis CFP10/ESAT-6 protein. PloS one, 8(1), e52848.***Scatter dot plot representation of the distribution of IgG levels to the peptide E5 and CE protein in TB patients (□) and negative control (▵).(Error bars: 95% confidence interval for mean).*### Serial measurements over time

When the same thing is measured over multiple time points, a good way to present the results is through a line plot as shown in the graph that follows. The plots can include confidence intervals and p-values.

The 95% confidence interval bars show the range within which the mean of that group is expected to lie with 95% confidence. Notice that similar error bars were presented earlier in this chapter (see section on bar charts to display the mean of a numeric variable), but these represented 'mean ± SD' (i.e. a range of the data). There is no consistency between and even within the same journals on what these lines mean, is it's important to include this information in the key or legend.

Confidence intervals are dependent on sample size (and are smaller for larger samples); standard deviations do not depend on sample size, they merely show where the values lie.

With plots of serial measurements, there is always a danger of multiple hypothesis testing. The following graph does not take this into account and compares groups at each time point separately. The plot does not report exact p-values either.

Plots such as these also lose the matching in the data - the same sample is measured over time. Unfortunately it's difficult to find a better way to present the results. You hope the authors have explored the individual trajectories over time prior to producing the final results.

Ref:

**Gronbaek et al. Family involvement in the treatment of childhood obesity: the Copenhagen approach.***Eur J Pediatr*(2009) 168:1437-1447### Regression

Regression was introduced in chapter 10, as a way of modelling the relationship between an outcome and one or more predictor/independent variables. Results of regression analyses can be displayed in different ways, the most appropriate will depend on the type of regression and the number of predictor/independent variables included in the model. Graphs showing results of regression models are quite rare as they are only possible when linear regression is used and either one numeric or one numeric and one categorical predictor/independent variables is included in the model. Two examples of this are given below:

Ref:

**Weiss, Günter, et al. "Associations between Cellular Immune Effector Function, Iron Metabolism, and Disease Activity in Patients with Chronic Hepatitis C Virus Infection." The Journal of Infectious Diseases, vol. 180, no. 5, 1999, pp. 1452-1458.**This graph displays the results of a linear regression model quantifying the association between serum ferratin levels (the outcome/dependent variable) and hepatic iron concentration (the predictor/independent variable). The graph includes a 'line of best fit' from the regression equation and the 95% confidence interval for this line. Pearson's correlation coefficient (r) is also superimposed on the plot.

The next example shows how a regression model with a single numeric predictor, and another binary predictor can be displayed in a graph.

Ref:

**Hosnijeh, Fatemeh Saberi, et al. "Changes in Lymphocyte Subsets in Workers Exposed to 2,3,7,8-Tetrachlorodibenzo-p-Dioxin (TCDD)." Occupational and Environmental Medicine, vol. 69, no. 11, 2012, pp. 781-786.***Figure 1 Correlation between B cell counts and TCDDmax levels among high- and low-exposed workers; both values are log-transformed; β=Linear regression slope estimate with p-value of the slope estimate. Results did not materially change after removing the outliers. TCDDmax indicates the 2,3,7,8-tetrachlorodibenzo-p-dioxin blood levels at the time of last exposure. This figure is only reproduced in colour in the online version.*If the regression model cannot be displayed using a scatterplot, either because there are too many predictors in the model or the outcome variable is not continuous, the results should be displayed in a table. Results contained within these tables may vary depending on the type of regression carried out but generally they will contain an estimate of the association between the given predictor and outcome (either the coefficient estimates or a transformation of these depending on the type of regression), confidence intervals for this association and a p-value testing the null hypothesis of no association.

It is common to see two tables of regression results in papers: the univariable results and multivariable (adjusted) results. Univariable regression models show the association between a variable and the outcome without adjusting for other characteristics. Displaying both univariable and multivariable tables in a paper is admissible as long as the univariable results are not used to choose the final model (see chapter 10).

Ref:

**Maan R, van der Meer AJ, Brouwer WP, Plompen EPC, Sonneveld MJ, Roomer R, et al. (2015) "ITPA Polymorphisms Are Associated with Hematological Side Effects during Antiviral Therapy for Chronic HCV Infection". PLoS ONE 10(10): e0139317**The table above contains results from two sets of analyses, univariable and multivariable, investigating the relationship between the decline in platelet count at week 4 (the outcome variable) and various characteristics. Coefficient estimates (Beta in this table) are given alongside confidence intervals and p-values to give estimates of the associations and a measure of the precision of these estimates. Note that after adjusting for other predictors, results can change drastically (for example, age was significantly related to platelet decline before adjusting for other factors).

When logistic regression has been carried out, it is common to see a transformed version of the coefficient estimates and confidence interval displayed to make results more easily interpretable and relatable to the outcome of interest.

- Summary
Although it may be argued that plots of summaries alone helps to convey the main message from the data, we do have to be aware that without the complete data it is impossible for the reader to judge whether the illusion given is a correct interpretation. We are dependent on the authors having selected appropriate summaries. An over simplistic picture may have been given. It may be preferable to display the raw data so that outliers can be identified and patterns discerned and summarized in the analyses only. Where possible, perhaps we should give the data alongside any results presented. The best option to choose will be informed at least in part by the number of data points to be displayed. For example, to give means and error bars, boxplots or regression lines to summarise groups of 10 or less is not generally acceptable, ideally there should be 20 or more per group to warrant using any form of summary measures. Any summaries should always, in plots as in analyses, be given with some measure of precision.