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Our survey data analyses so far have focused on tool usage for specific research activities (e.g. GitHub and others: data sharing, Who is using altmetrics tools, The number games). As a next step, we want to explore which tool combinations occur together in research workflows more often than would be expected by chance. This will also facilitate identification of full research workflows, and subsequent empirical testing of our hypothetical workflows against reality.

Checking which tools occur together more often than expected by chance is not as simple as looking which tools are most often mentioned together. For example, even if two tools are not used by many people, they might still occur together in people’s workflows more often than expected based on their relatively low overall usage. Conversely, take two tools that each are used by many people: stochastically, a sizable proportion of those people will be shown to use both of them, but this might still be due to chance alone.

Thus, to determine whether the number of people that use two tools together is significantly higher than can be expected by chance, we have to look at the expected co-use of these tools given the number of people that use either of them. This can be compared to the classic example in statistics of taking colored balls out of an urn without replacement: if an urn contains 100 balls (= the population) of which 60 are red (= people in that population who use tool A), and from these 100 balls a sample of 10 balls is taken (= people in the population who use tool B), how many of these 10 balls would be red (=people who use both tool A and B)? This will vary with each try, of course, but when you repeat the experiment many times, the most frequently occurring number of red balls in the sample will be 6. The stochastic distribution in this situation is the hypergeometric distribution.

For any possible number x of red balls in the sample (i.e. 1-10), the probability of result x occurring at any given try can be calculated with the hypergeometric probability function. The cumulative hypergeometric probability function gives the probability that the number of red balls in the sample is x *or higher*. This probability is the p-value of the hypergeometric test (identical to the one-tailed Fisher test), and can be used to assess whether an observed result (e.g. 9 red balls in the sample) is significantly higher than expected by chance. In a single experiment as described above, a p-value of less than 0.05 is commonly considered significant.

In our example, the probability of getting *at least* 9 red balls in the sample is 0.039 (Figure 2). Going back to our survey data, this translates to the probability that in a population of 100 people, of which 60 people use tool A and 10 people use tool B, 9 or more people use *both *tools.

In applying the hypergeometric test to our survey data, some additional considerations come into play.

**Population size
**First, for each combination of two tools, what should be taken as total population size (i.e. the 100 balls/100 people in the example above)? It might seem intuitive that that population is the total number of respondents (20,663 for the survey as a whole). However, it is actually better to use only the number of respondents who answered the survey questions where tools A and B occurred as answers.

People who didn’t answer both question cannot possibly have indicated using both tools A and B. In addition, the probability that at least x people are found to use tools A and B together is lower in a large total population than in a small population. This means that the larger the population, the smaller the number of respondents using both tools needs to be for that number to be considered significant. Thus, excluding people that did not answer both questions (and thereby looking at a smaller population) sets the bar higher for two tools to be considered preferentially used together.

**Choosing the p-value threshold**

The other consideration in applying the hypergeometric test to our survey data is what p-value to use as a cut-off point for significance. As said above, in a single experiment, a result with a p-value lower than 0.05 is commonly considered significant. However, with multiple comparisons (in this case: when a large number of tool combinations is tested in the same dataset), keeping the same p-value will result in an increased number of false-positive results (in this case: tools incorrectly identified as preferentially used together).

The reason is that a p-value of 0.05 indicates there is a 5% chance the observed result is due to chance. With many observations, there will be inevitably be more results that may seem positive, but are in reality due to chance.

One possible solution to this problem is to divide the p-value threshold by the number of tests carried out simultaneously. This is called the Bonferroni correction. In our case, where we looked at 119 tools (7 preset answer options for 17 survey questions) and thus at 7,021 unique tool combinations, this results in a p-value threshold of 0.0000071.

Finally, when we not only want to look at tools used *more often* together than expected by chance, but also at tools used *less often* together than expected, we are performing a 2-tailed, rather than a 1-tailed test. This means we need to halve the p-value used to determine significance, resulting in a p-value threshold of 0.0000036.

**Ready, set, …**

Having made the decisions above, we are now ready to apply the hypergeometric test to our survey data. For this, we need to know for each tool combination (e.g. tool A and B, mentioned as answer options in survey questions X and Y, respectively):

a) the number of people that indicate using tool A

b) the number of people that indicate using tool B

c) the number of people that indicate using both tool A and B

d) the number of people that answered both survey questions X and Y (i.e. indicated using at least one tool (including ‘others’) for activity X and one for activity Y).

These numbers were extracted from the cleaned survey data either by filtering in Excel (a,b (12 MB), d (7 MB)) or through an R-script (c, written by Roel Hogervorst during the Mozilla Science Sprint.

The cumulative probability function was calculated in Excel (values and calculations) using the following formulas:

**=1-HYPGEOM.DIST((c-1),a,b,d,TRUE)**

(to check for tool combination used together *more often* than expected by chance)

and

**=HYPGEOM.DIST(c,a,b,d,TRUE)**

(to check for tool combination used together *less often* than expected by chance)

Bonferroni correction was applied to the resulting p-values as described above and conditional formatting was used to color the cells. All cells with a p-value less than 0.0000036 were colored green or red, for tools used *more* or *less* often together than expected by chance, respectively.

The results were combined into a heatmap with green-, red- and non-colored cells (Fig 4), which can also be found as first tab in the Excel-files (values & calculations).

**Pretty colors! Now what?**

While this post focused on methodological aspects of identifying relevant tool combinations, in future posts we will show how the results can be used to identify real-life research workflows. Which tools really love each other, and what does that mean for the way researchers (can) work?

*Many thanks to Bastian Greshake for his helpful advice and reading of a draft version of this blogpost. All errors in assumptions and execution of the statistics remain ours, of course *