The following plot shows the results of the regression on the original data from the previous chapter. The horizontal axis represents the age of the prospective mother and the vertical axis represents the associated success rates when the prospective mother's own eggs are used in ART.

There are several curves shown on the plot, the regression line and two sets of confidence bounds.

• The middle line is the regression line computed in the previous regression analysis.
• Also shown are two kinds of confidence bounds.
  • The inner bounds show 95.0% confidence limits for the mean of the variable Success Rate at given values of the variable Age.
  • The outer bounds show 95.0% confidence limits for a individual value of the variable Success Rate at given values of the variable Age.

  The Studentized Residuals plot shows the residuals. It shows a curvilinear pattern, which indicates that the model could be improved. It seems that there is a curvilinear (perhaps quadratic) effect that we have omitted in the model.

When you see an obvious effect in your residual plot, consider adding that effect to your regression model. You may consider analyzing the data again by adding a quadratic effect (age²) to the regression model.

Carefully examine the following plot: it shows a very different relationship. In the light of your previous analysis, what do you make of this situation?

The success rate is measured just as before, but now, the eggs used do not belong to the prospective mother, but come from donors.

Therefore, we might expect no relationship with age. The graph seems to show a positive relationship, however. This is hard to understand. In fact, without more information it is impossible to explain.

We are not sure exactly why this regression shows a positive relationship with age. It could be a statistical accident, even though it is a statistically significant result (recall that p-values are values of probability and we are not guaranteed that the relationship is a true one, only that there is high probability that it is true).

The regression analysis table is as follows:

Equation: SuccessRate = 16.365 + 0.475*age
-----------------------------------------------------------------------
Coefficient     Estimate        Std. Error      t-value         P-value
-----------------------------------------------------------------------
Intercept       16.365          3.07799         5.31679         1.0E-4
Slope           0.475           0.0816526       5.81733         1.0E-4
-----------------------------------------------------------------------
Correlation = 0.7716
R-squared = 59.54%
Std. error of est. = 2.94402

The residual plots shows no obvious trend:

Think about the concept of statistical significance...accidents do, of course, happen, and data does fluctuate. Otherwise, we would have little need for statistics.

One possible explanation may be related to the methods used in assisted reproductive therapy. When the prospective mother is toward the end of her child-bearing years, the number of implanted eggs is increased in ART. Perhaps this is the reason for the small positive slope seen in the data: a tendency to overcompensate for a woman's age by implanting a higher number of eggs. This is only a theory, and the relationship may be a statistical anomaly: obviously more information is needed to develop a confident answer to the question.