1. Case introduction
To study the effects of temperature (°C), time (min), and catalyst content (%) on the conversion rate of a certain substance, we are currently using an orthogonal experimental design to conduct experiments to explore whether the impact of the three factors on the conversion rate is significant, and find Optimum conditions for maximum conversion. The factor level table and the results of the orthogonal test are shown in the following table respectively:
Table 1 Factors and levels
Table 2 Orthogonal design and test results
2. Problem Analysis
In this case, an orthogonal test design is used for the test. The purpose of the analysis is to explore the significance of the three factors and to find the best conditions to maximize the conversion rate. To analyze the results of the orthogonal test, you can use the range analysis method or the variance analysis method. Although the range analysis is simple and easy to understand, it does not consider whether each factor has a significant impact on the test results. Therefore, this case can be studied using analysis of variance. There are 3 factors in the study, specifically three-factor analysis of variance.
3. Software operation and result interpretation
(1) Software operation
Upload the data to the SPSSAU system, in the [Advanced Method] module, select [Three-factor ANOVA], drag "Conversion Rate" to the Y analysis box on the right; drag and drop the 3 factors to the X analysis box on the right Medium; There are many methods for multiple comparisons after the event, among which the LSD method is more commonly used; click "Start Analysis" and the operation is as shown in the figure below:
figure 1
(2) Result analysis
① Factor significance judgment
The SPSSAU output three-way ANOVA results are as follows:
figure 2
Analysis of the above table shows that the influence of temperature on the conversion rate is significant at the 0.05 level (F=34.3333, p=0.0283<0.05), which is the best factor; the catalyst content is significant at the 0.1 level (F=13, p= 0.0714<0.1); the effect of time on the conversion rate was not significant (F=6.3333, p=0.1364>0.1). Therefore, the primary and secondary order of the influence of the three factors was found to be: " temperature>catalyst content>time ".
If you want to get the best condition with the highest conversion rate, you can analyze it through post-hoc multiple comparison results.
② Best condition judgment
SPSSAU outputs post hoc multiple comparison results as follows:
image 3
Figure 4
Figure 5
According to Figure 3, analyze the post-hoc multiple comparison results of factor A temperature: when the temperature is 80°C and 90°C, the corresponding p value of the t-test is the smallest at this time, indicating that the difference between the two is the most significant. Specifically, check the mean difference value of -20, indicating that the average conversion rate corresponding to 80°C is 20 lower than the average conversion rate corresponding to 90°C, that is, the corresponding conversion rate is the highest when the temperature is 90°C.
Analyzing the multiple comparison results of factor B time and factor C catalyst content in Figure 4 and Figure 5 in the same way, it can be known that the corresponding conversion rate is the highest when the time is 120 minutes and the catalyst content is 6%.
To sum up, the combination of conditions with the highest conversion rate is: 3 levels of factor A, 2 levels of factor B and 2 levels of factor C, that is, temperature 90°C, time 120min, catalyst content 6%.
4. Conclusion
This case uses three-factor analysis of variance to study the influence of three factors on the conversion rate. The analysis shows that temperature is the main factor affecting the conversion rate (F=34.3333, p=0.0283<0.05); the second is the catalyst content, and the effect of time on the conversion rate The impact did not appear to be significant. Through multiple comparisons after the fact, it was obtained that the combination of conditions to achieve the highest conversion rate was temperature of 90°C, time of 120min, and catalyst content of 6%.
5. Knowledge Tips
Why is it sometimes impossible to do multi-factor ANOVA?
Orthogonal table and multi-factor ANOVA are completely independent, so sometimes the orthogonal table comes out but multi-factor ANOVA cannot be performed because the number of experiments is too small and the degree of freedom is insufficient to perform multi-factor ANOVA. For example, the orthogonal table L9.3.4 is an orthogonal table with 4 factors and 3 levels. If multi-factor analysis of variance is required, at least the required number of degrees of freedom needs to be greater than: 4*(3-1)+1=9, then at least 10 experiments are required to perform multi-factor analysis of variance with an orthogonal table of 4 factors and 3 levels . There are two solutions, one is to choose an orthogonal table with a higher number of experiments; the other is to do at least one more experiment by yourself (and the experimental combination cannot be the same as the existing combination in the orthogonal table).