Dentists' Working Patterns, Motivation and Morale - 2022/23

Multiple Linear Regression Analysis

Linear regression analysis attempts to model the relationship between a dependent variable y (in the case of this report, the ‘motivation index’) and an independent (explanatory) variable denoted x (such as average weekly hours or NHS/HS share). Multiple linear regression considers more than one explanatory variable at a time (x₁, x₂, x₃…) and quantifies the strength of relationships between them (recorded as parameter estimates) and the dependent variable y.

Unlike other results in the report, the multiple linear regression analysis (and logistic regression provided in the accompanying Multivariate Analysis csv file) is not weighted. Normally in regression analysis it is assumed that the standard deviation of the error term is constant over all values of the explanatory variables. Weighting allows certain observations to be assigned more weight in the regression analysis (thereby having more influence on the calculated parameter estimates) because it is believed that they are more accurate. Without weighting all the observations are treated equally, which is the preferred option for this report.

The ‘goodness of fit’ of a multiple linear regression model can be assessed by considering the adjusted R-squared value produced in the analytical output. The adjusted R-squared value measures the proportion of the variation in the dependent variable accounted for by the explanatory variables. The values of adjusted R-squared range from 0 to 1, with a value of 1 indicating that the regression line perfectly fits the data. The results in the report are generally lower than 0.2, which would traditionally be seen as low in many objective prediction models. However, in prediction models used in psychology, the R-squared values tend to be lower. This is because the goal is to determine which variables are statistically significant and how they relate to changes in the response variable, meaning that the adjusted R-squared value becomes less important.

Assumption Testing of Linear Regression Models^[1]

There are four principal assumptions supporting the use of linear regression models for purposes of inference:

1. Linearity and additivity of the relationship between the dependent and independent variables
2. Statistical independence of errors
3. Constant variance of the errors
4. Normality of the error distribution

Violations of linearity or additivity are extremely serious: if a linear model is fitted to data which are non-linearly or non-additivity related, then the predictions are likely to be in serious error. In multiple regression models, non-linearity or non-additivity may be tested by systematic patterns in plots of the residuals versus individual explanatory variables. The points should be symmetrically distributed around the horizontal line with a roughly constant variance. In terms of statistical independence of errors, the residuals should be randomly and symmetrically distributed around zero under all conditions. To satisfy constant variance of errors, the residuals should not get systematically larger in any one direction of the explanatory variables by a significant amount.

Violations of normality create problems for determining whether model coefficients are significantly different from zero and for calculating confidence intervals for forecasts. Sometimes the error distribution is "skewed" by the presence of a few large outliers. Since parameter estimation is based on the minimization of squared error, a few extreme observations can exert a disproportionate influence on parameter estimates. Calculation of confidence intervals and various significance tests for coefficients are all based on the assumptions of normally distributed errors. If the error distribution is significantly non-normal, confidence intervals may be too wide or too narrow.

One of the best tests for normally distributed errors is a normal quantile plot of the residuals. This is a plot of the fractiles of error distribution versus the fractiles of a normal distribution having the same mean and variance. If the distribution is normal, the points on such a plot should fall close to the diagonal reference line.

^[1] The majority of the material used in this section was taken from ‘Statistical forecasting: notes on regression and time series analysis’ website, Faqua School of Business, Duke University: http://people.duke.edu/~rnau/testing.htm

Detailed Regression Results and Assumption Testing

As discussed in Annex D, several analytical assumptions have been made to investigate motivation in dentists. Many of these assumptions relate to the calculation of the average ‘motivation index’ and its modelling by linear regression. This Annex gives the linear regression results for 2022/23 for England in more detail and tests the assumptions around the use of the multivariate analysis for the ‘motivation index’.

England

Table E1 shows multiple linear regression results by dental type for the average ‘motivation index’. This table also shows the parameter estimate results together with the p-value for each and the adjusted R-squared results. The intercept represents value of the ‘motivation index’ when all explanatory variables are zero.

Table E1: Parameter estimates and p-values for ‘motivation index’ by dental type using multiple linear regression, England, 2022/23

The positive or negative values of the variable results indicate the relationship between each variable and the average ‘motivation index’.  For both sets of dentists an increase in annual leave correlates with an increase in motivation, whereas increases in NHS share (%) have the opposite effect and lower motivation.

This type of analysis allows the individual relationships between each measured variable and the ‘motivation index’ to be assessed. For example, the statistical model predicts that if all other working patterns remained unchanged, but dentists switched from all private to entirely NHS work (from 0% to 100% NHS share) the ‘motivation index’ of Providing-Performers would decrease by 24.4 percentage points and by 20.5 for Associate dentists.

Whilst the weekly hours of work results are not significant for either Providing-Performer or Associate dentists, removing part-time dentists (<35 weekly hours) from the analysis changes the coefficient scores from -0.13 to -0.36 for Providing-Performers and from 0.06 to -0.61 for Associates with both results becoming significant as shown in table E2 below. This suggest that choosing to work part-time limits the negative relationship between motivation and weekly hours of work, which may be one reason that some dentists make this work-life choice.

Finally, it is important to note that whilst regression analysis provides evidence for the existence of relationships between variables, it does not provide measures of causality. In other words, although there may well be a relationship between, for example, weekly hours of work and motivation, it is not possible to determine if longer working hours demotivate staff or whether demotivated staff tend to work longer hours.

Based on the results it is possible to predict the ‘motivation index’ for both dental types. For example, for Providing-Performers this is given as:

43.789 - (0.126 x WH) - (0.244 x NHS%) + (0.049 x Clinical%) + (1.825 x Leave) - (0.277 x Age)

Residual plots for Providing-Performers are shown in Figure E1.

Figure E1: Providing-Performer dentists, plot of residuals for average ‘motivation index’ versus individual independent variables, England, 2022/23

Overall, the residual plots support the first three assumptions of multiple linear regression for average ‘motivation index.

Figure E2 shows the normal quantile plot of the residuals for Providing-Performers.

Figure E2: Providing-Performer dentists, normal quantile plot of residuals for average ‘motivation index’, England, 2022/23

The result is a linear pattern plot with no significant departure from normality.

Similar plots to the above are shown for Associate dentists in Figures E3 and E4.

Figure E3: Associate dentists, plot of residuals for average ‘motivation index’ versus individual independent variables, England, 2022/23

Figure E4: Associate dentists, normal quantile plot of residuals for average ‘motivation index’, England, 2022/23

The residual plots for Associate dentists appear to support the first three assumptions for multiple linear regression of the average ‘motivation index’ as does the almost linear quantile plot.

Different Dental Populations

Table E2 compares multivariate results for the average ‘motivation index’ for Providing-Performer and Associate dentists.  The first rows ('All') repeat results from table E1 above, with subsequent rows showing results for different cohorts of dentists. This table also shows the parameter estimate results together with the p-value for each and the adjusted R-squared results.

Table E2: Parameter estimates and p-values for ‘motivation index’ by dental type and split populations using multiple linear regression, England, 2022/23

There are some interesting results when dentists are split into different cohorts. For example, both Providing-Performer and Associate dentists who work full-time (≥35 hours) exhibit a statistically significant negative relationship between ‘motivation index’ and weekly hours of work, whilst their part-time colleagues (<35 hours) do not^[2]. 

It is also noticeable, that unlike any other variable, NHS share (%) exhibits a statistically significant and negative relationship with ‘motivation index’ for the majority of cohorts of dentists.