What is It?

Video 1 defines the response surface methodology, RSM with examples.

Video 1. Introduction to Response Surface Methodology

Video 2 demonstrates response surface methodology analysis with examples in Minitab.

Video 2. Introduction to Response Surface Methodology Explained with Analysis Example

Response Surface Methodology, RSM (also known as Response Surface Modeling) is a technique to optimize the response(s) when two or more quantitative factors are involved. The dependent variables are known as responses, and the independent variables or factors are primarily known as the predictor variables in response surface methodology. While p-values are used for a particular point such as to test the hypothesis of “whether the 70-degree Fahrenheit is the most comfortable temperature or not,” the response surface is useful in determining a range of temperatures for the same comfort level. As maintaining the temperature exactly at a 70-degree could be very expensive, maintaining the temperatures within a range is often desired for a cost-effective solution. Moreover, keeping very cool in summer or very hot in winter would be very wasteful. Response Surface Methodology, RSM, is very useful to optimize variables/factors more practically as compared to just the statistical significance test for a particular point (point estimate is the statistical jargon). For example, to optimize humidity and temperature for the best comfort, the response surface is plotted in Figure 1. Human comfort is measured on a scale between 0 to 1o, where 10 is the most comfortable.

Figure 1. Response Surface Plot of Comfort vs Humidity and Temperature

While the response surface is visually appealing and provides a quick meaningful overview of the relationship, a contour plot is easier to understand with respect to the optimized values for the independent variables, for which the same level of comfort (response or dependent variable) can be achieved. For example, the contour plot in Figure 2 shows that statistically the same level of comfort can be achieved for the same color reason in the plot. For example, the middle dark oval region of the contour plot in Figure 2 represents the comfort level over 7.5 can be achieved for the temperature approximately between 65- and 78-degrees Fahrenheit and between 25 and 70 percent of relative humidity. Similarly, the contour plot can be used to find the ranges of temperature and humidity to achieve the same level of comfort (response, y).

Figure 2. Contour Plot of Comfort vs Humidity and Temperature

In this particular example for human comfort study, while maximizing comfort is the goal, minimizing the response (dependent variable) would also be desired for situations such as human discomfort study provided in Figure 3 and Figure 4. While comfort is not simply the opposite of discomfort, discomfort study for sure wants to minimize the response (discomfort) optimizing the independent variables (temperature and humidity in this case). Therefore, the response surface methodology could either be used for maximizing or minimizing the responses. These maximum responses or minimum responses are known as the stationary point. If there are optimized points that exist for the independent variables, the partial derivatives for these points will be equal to zero (Equation 1) as it can be understood from both maximum and minimum responses in Figure 1 & Figure 3.

Equation 1

Figure 3. Response Surface Plot of Discomfort vs Humidity and Temperature

Figure 4. Contour Plot of Discomfort vs Humidity and Temperature

In addition to the maximum and minimum response stationary points, there is a third type of optimization called the saddle point could also exist as a stationary point. Similar to a horse saddle or bicycle saddle, this optimization could be visualized through Figure 5 and Figure 6.

Figure 5. Response Surface Plot of Saddle Point Type Optimization

Figure 6. Contour Plot of Saddle Point Type of Optimization