One-Way Analysis    Targeting    Two-Way Analysis    Regression

Topics on this page:

 Uses of Regression      Basic Equation         Interaction      Graphing        Creating dichotomous variables                    
  Building a Regression Model to control for Confounders

Regression is used:

  with a continuous outcome (or dependent) variable

  with continuous and categorical (usually 0 and 1) determining (or independent) variables, in order to:

                        look for confounding

                        look for interactions

              confirm findings from tabulation, and vice versa

This section introduces simple regression using two independent variables. It will provide the fundamentals of regression analysis. Regression is especially powerful for exploring more than two independent variables, and it will be most useful for multi-way analysis in the following module.

The Basics of a Regression Equation

The following graph shows a simple (one independent variable) regression equation and where each of the variables is represented.

Developing a Regression Equation

To start, it is a good idea to have a look at the associations of the independent variables and the outcome, using scatterplotting, that relates to your research question. Here is the question we will use for analysis in this section:

• Are the variables normally distribued?
  We have already established in the one-way section that these variables (cult, irr99) are not normally distributed, but using log transformations allow us to use both variables in regression.  In order to perform a regression analysis, the variables being used must be normally distributed, or in other words,
  have a bell-curve shaped histogram.

• What are the one-way associations?
  As a first step in the analysis, use scatterplots to show one-way associations between independent variables
  (irrigation & cultivation) and outcome variables (food sufficiency (# of months of food stores) or meals per day).

                                 Here is an example of a scatterplot:

There is a positive association with the level of food sufficiency and the area of land under cultivation for the household.
This is shown by the slope of the fitted line (we will check in a moment to see if this is significant). This positive slope is reasonable, considering that generally the more land under cultivation the greater the amount of food produced.

Here, the fitted line shows the association between the outcome, food sufficiency, and the independent variable, area under cultivation (it is calculated by minimizing the square of the value of the sum of all the distances between the observation and the line -- hence, the least squares line). The line fits all of the points marked
on the graph so that the deviation squared from the line for all points combined is the smallest possible for that set of points. The deviations from the line are used to predict other features of the relationship between outcome and independent variables. One of these is the coefficient of determination (R-squared), which tells the amount of variability in the outcome that is due to the independent variable (percent of change in food sufficiency due to the area cultivated). An r-square of
0.0031 is telling us that very little of the variability in food sufficiency is explained by cultivation.

• What is the associated regression equation using cultivation?
  (remember: the dependent variable on the left-hand side and the independent on the right-hand side):
  Level of Food Sufficiency = A + B (log area cultivated)
                          or with real numbers:
  Level of Food Sufficiency = 3.114 + 0.424 (log area cultivated)

          Significance for cultivation (p-value) = 0.300, thus non-significant.

The same analysis can be done using irrigation:

The association seen here is also positive, but much stronger than the association found using area cultivated. The r-square is much larger (0.0855), but to test the significance of the association we must again use regression.

• What is the regression equation using irrigation?
  Level of Food Sufficiency = A + B ( log area irrigated)|
                       or with real numbers:
  Level of Food Sufficiency = 3.477 + 1.029(log area irrigated)
  
  Significance for irrigation (p-value) = 0.024, thus significant.

Not only is the r-square larger and the equation statistically significant (p-value < .05), the coefficient for area irrigated is
more than two times larger than the coefficient for cultivation (0.424) at 1.029, meaning that for every one unit rise in area irrigated (hectares) a corresponding 1.029 month increase in food sufficiency is estimated.

• What is the regression equation using both cultivation and irrigation?
  Food Sufficiency = A + B(logcult) + C(logirr99)
                 or using actual numbers:
  Food Sufficiency = 3.468 + 0.404(logcult) + 0.678(logirr99)
  
  Significance for cultivation: 0.743, thus insignificant
  
  Significance for irrigation: 0.579, thus insignificant

These results are interesting because the regression coefficient for irrigation is now an insignificant predictor of food sufficiency when controlling for cultivation. The coefficient for irrigation was also greatly reduced from 1.029 to 0.678 when cultivation was added to the model.  Before we can come to any conclusions from our findings, we first need to test for interaction between cultivation and irrigation.

• Does interaction exist?
  For program design, it is important to determine whether significant interactions are occurring, as this can determine whether you need two interventions to get an effect, or whether one or another may be expected to have an impact. Put the interaction term (variable) in the equation with the original variables. If it is significant (usually p=<.05, unless it is a highly interesting variable) keep it in; if not, drop it and rerun the regression equation without it. The coefficients of the variables will be affected by the interaction term because of multi-collinearity.   Testing for interaction is a step that must be taken, because without testing for it your results may be invalid if interaction does indeed exist amongst the variables.
  
• Once the interaction variable has been created, and we can now proceed with our regression exercise testing for interaction.
  

• What does our complete regression equation look like with the interaction variable?
  Food Sufficiency = A + B(logcult) + C(logirr99) + D(logcult*logirr99)
                             or using actual numbers:
  Food Sufficiency = 3.364 – 0.0079(logcult)+ 0.316(logirr99)+ 1.872(logcult*logirr99)
  Significance for cultivation: 0.995, thus insignificant
  
  Significance for irrigation: 0.797, thus insignificant
  
  Significance for interaction: 0.059, thus borderline significant
  These results indicate that there is significant interaction occurring between the area of land irrigated and cultivated. 
  
• How can the results be graphed?
  Creating dummy variables (dichotomous) for the continuous variables cult and irr99 allow us to plot the regression and look at the graph for interactions.  When plotting the potential interaction results in a graph with lines of noticeably different slopes, then interaction does indeed exist.

  
  
  This interaction could be catagorized as LESS THAN ADDITIVE: When the results are less than additive, neither factor shows an effect when improved alone because both are needed to see an improvement.