| Freshness | Box Office ($M) | ||||
|---|---|---|---|---|---|
| Valid | 31 | 31 | |||
| Missing | 0 | 0 | |||
| Mean | 0.272 | 78.206 | |||
| Std. Deviation | 0.201 | 49.593 | |||
| Minimum | 0.000 | 2.300 | |||
| Maximum | 0.790 | 162.000 | |||
In the Bayesian correlation analysis, we start by specifying which variables we want to include in our analysis. JASP will then output the table below, which shows the correlations and Bayes factor for all pairs of specified variables. Since we only have 2 variables, we only see a single correlation (since we only have a single pair).
(3) The observed correlation coefficient r is -0.029.
| Variable | Freshness | Box Office ($M) | |||||
|---|---|---|---|---|---|---|---|
| 1. Freshness | Pearson's r | — | |||||
| BF₁₀ | — | ||||||
| 2. Box Office ($M) | Pearson's r | -0.029 | — | ||||
| BF₁₀ | 0.226 | — | |||||
In order to take a closer look at a specific pair of variables, we can go to the submenu "Plot Individual Pairs". Here we can drag variables into the variable box to make pairs of variable, which are then analyzed closer. For instance, below we have a scatter plot and prior/posterior plot for the specific combination of "Freshness" and "Box Office".
(4) The plot below the scatterplot displays the prior and posterior distribution for the correlation coefficient rho (i.e., the population parameter). The prior distribution is uniform between -1 and 1 (this happens when you set the prior width to 1), which means that this model "bets" equally much on all possible values of rho. When we update the prior distribution with the observed data, we obtain the posterior distribution. As you can see below, the observed correlation of -0.029 caused an increase in plausibility for values around 0 (because these values predicted the data well). The posterior median is equal to -0.026, and the 95% credible interval ranges from -0.363 to 0.314. This means that under this model, there is a 95% probability (CI) that the true value of rho lies between -0.363 and 0.314, and that there is a 50% probability (median) that the true value of rho is smaller than -0.026.
(5) The prior probability of rho = 0 is 0.5 (i.e., it is the height of the prior distribution at 0). The posterior probability of rho = 0 is around 2.2 (i.e., it is the height of the posterior distribution at 0). In the plot below, these points are indicated by the gray dots. Because the posterior probability is higher than the prior probability, this means that the data have increased the plausbility that the population parameter rho is equal to 0 (i.e., the predictive updating factor for that specific value is greater than 1). In fact, the ratio of these two values is equal to the Bayes factor: 2.2 / 0.5 = 4.4. In other words, the Bayes factor is equal to the predictive updating factor for the "test-value" (in this case 0, because we want to test if there is an association or not).
When we look at the Bayes factor, we compare the alternative model to the null model. The alternative model is specified by the prior distribution: in this case, the alternative model "bets" equally much on all possible values of rho. The null model only bets on a single value: 0 (if we were to make a distribution plot of what this bet looks like, it would be a single "spike" at 0). The Bayes factor is the ratio between how well the alternative model predicted the data, and how well the null model predicted the data. Using the betting analogy, this is the ratio between the models' "winnings". The alternative model bet on all values of rho, which means that it also bet on some values that predicted the data very poorly - so on average, it did not do very well. The null modle went "all in" on the value 0 - this value predicted the data very well (because the oberved correlation is very close to 0), so it did very well.
(6) The Bayes factor in favor of the null hypothesis (BF01) is equal to 4.429 and the Bayes factor in favor of the alternative hypothesis (BF10) is equal to 1 / 4.429 = 0.226. This means that the data are 4.429 times more likely to have occured under the null hypothesis, than under the alternative hypothesis. In other words, the null model had 4.429 as much "winnings" as the alternative model. We can view this as moderate evidence in favor of the null hypothesis.
In order to investigate the one-sided hypothesis that there is a positive relationship between the two variables, we can use a one-sided positive alternative hypothesis. This hypothesis only predicts positive values for the correlation. This is reflected by the prior distribution (it has no mass for negative values of the correlation). We use the default prior distribution shape, which is the uniform distribution (stretched beta width = 1).
We return to the Bayes factor, but this time our alternative model is different. This time, the alternative model only bets on positive values of rho (as compared to betting on all values, which we did in the two-sided test). The null model is still placing the same bet: all in on 0.
(7) Even though the alternative model is now making a more specific prediction (i.e., a positive correlation), it still bet on pretty implausible values of rho. As you can see from the prior and posterior plot, values of rho greater than +- 0.3 suffered a decrease in plausibility as a result of obesrving the data (i.e., the prior probability is higher than the posterior probability). And the alternative model placed bets on all those values, while the null model bet on a value that predicted the data very well. As a result, the one-sided Bayes factor is still in favor of the null hypothesis: BF0+ = 5.02. The data are 5.02 times more likely under the null hypothesis than under the alternative hypothesis. It is slightly higher than the two-sided Bayes factor.
(8) Next, we can investigate to what extent our result might differ for different settings of the prior width. We can do this by looking at the Robustness plot below. On the x-axis we have different values for the width, and the y-axis lists the corresponding Bayes factor. For instance, for a prior width of 2, the Bayes factor is about 1 / 9 = 0.11 in favor of the alternative hypothesis (or 9 in favor of the null hypothesis). We can see that in all cases we will obtain evidence in favor of the null hypothesis. As you can see, the Bayes factor goes towards 1 as the prior width decreases. As the prior width decreases, the prior distribution becomes more and more centered around 0. Since the prior distribution reflects the predictions made by the alternative hypothesis, this means that the alternative hypothesis starts betting more on values close to 0. The most narrow prior is basically a "spike" at 0, which means that the alternative hypothesis goes "all in" on the value 0 - sound familiar? Because now both the alternative and null hypotheses make exactly the same bet, and so the ratio of their winnings is 1.
| Variable | Freshness | Box Office ($M) | |||||
|---|---|---|---|---|---|---|---|
| 1. Freshness | Pearson's r | — | |||||
| BF₊₀ | — | ||||||
| 2. Box Office ($M) | Pearson's r | -0.029 | — | ||||
| BF₊₀ | 0.199 | — | |||||
| Note. For all tests, the alternative hypothesis specifies that the correlation is positive. | |||||||
(9) Lastly, we can look at negative one-sided hypothesis test, to test Adam's theory. Now, the alternative hyothesis only predicts negative correlation values. This alternative hypothesis also bets on values of rho that predicted the data pretty poorly (i.e., values lower than -0.4 received a penalty in plausility as a result of the data). This means that the null hypothesis agains outperformed the alternative hypothesis. The Bayes factor BF0- is equal to 3.962 (or, BF-0 is equal to 1/ 3.962 = 0.252). This means there is moderate evidence in favor of the null hypothesis that there is no relation between the two variables. We would therefore not really support Adam in his claim that they are negatively correlated.
| Variable | Freshness | Box Office ($M) | |||||
|---|---|---|---|---|---|---|---|
| 1. Freshness | Pearson's r | — | |||||
| BF₋₀ | — | ||||||
| 2. Box Office ($M) | Pearson's r | -0.029 | — | ||||
| BF₋₀ | 0.252 | — | |||||
| Note. For all tests, the alternative hypothesis specifies that the correlation is negative. | |||||||