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【中圖分类号】 E920.8 【文献标识码】 A 【文章编号】 2236-1879（2017）14-0298-02
1Introduction
A designed experiment was conducted into the e ects of certain toxic agents on 48 samples of animals. The given dataset contains survival times of 48 animals. We are interested in examining the in uence of two factors： Poison type and Treatment on survival times and rates of 48 animals. There are three poison types（I， II， III） and four treatments（A， B， C， D）， a group of four animals were randomly allocated to each combination of two factors， which means four animal samples of each of the three poison types were sent to each treatment. In this report， we conduct two statistical analyses of survival times and rates respectively. We then compare two analyses and summarise a conclusion to this experiment in the last section.
2Statistical analysis
In this experiment， there are two qualitative explanatory variables： Poison type and Treatment. Since there is also the possibility that the e ect of changing the treatment depends on the poison type or， vice versa. This e ect between two classi cations is known as interaction e ect， therefore， we perform a two-way analysis of variance with consideration of interactions between two factors for statistical analyses.
2.1Statistical analysis of survival times
In the statistical analysis of survival times， there is one quantitative response variable： survival times of 48 sampled animals. Firstly， we are interested in whether interactions between Poison type and Treatment on survival times is signi cant or not.
Representing the data： we illustrate interactions between Poison type and Treatment in the interaction plots below （Figure 1）. As is shown in Figure 1， the lines on each plot are roughly parallel. But for some combina-tions of two factors， there are interaction e ects on survival times.
Figure 1. Interaction Plots
From Figure 1， we can observe that Poison type I and Treatment B have longest mean survival times， while Poison type III and Treatment A have the shortest. In order to test whether the overall interactions between e ects of Poison type and Treatment is signi cant or not， we need to conduct a further two-way analysis of variance with interactions for 48 animals' sur-vival times against Poison type and Treatment.
The two-way analysis with interactions gives the following output in R：
Response： Survival times（hours）
Table 1. ANOVA Table The fth column （Pr（>F）） in the above table contains p-value， which is the signi cant probability of each test statistic. The p-value of Interaction term（0.1123） is more than 1%， thus we retain our null hypothesis which means there is no evidence of overall interactions between the e ects of two factors in the interaction test. However， p-values of Poison type and Treatment are both far less than 0.001. Thus there is a strong evidence that main e ects of Poison type and Treatment in uence the survival times： average survival times are not same across three poison types and four treatment groups. The e ects of two factors on survival times are individually signi cant.
Besides， a normal plot of the residuals based on above two-way analysis of variance（right gure below） shows the assumption that the survival times are normally distributed is not quite satis ed since there exists many outliers
far away from the straight line in Normal Q-Q plot. And a plot of residuals against the tted values（left gure below） indicates that variance increases obviously with expected value. Thus the assumption of constant variance is also unsatis ed， and since both assumptions are invalid， this statistical analysis based on Normal linear model is not quite reasonable.
Figure 2. Diagnostic Plots
2.2Statistical analysis of rates
The designed experiment is also interested in analysing the rates of animals to death， which are de ned as reciprocal values of the survival times with
unit as hours1 ： Now we want to examine the e ects of di erences in Poison type and Treatment as well as interactions between two factors on the
rates to death of animals.
We illustrate interactions between Poison type and Treatment in the interaction plots below（Figure 3）. Since the lines on each plot are roughly parallel， there is little evidence of interactions between the e ects of two factors. But again for this analysis， both Poison type and Treatment have substantial e ects on the response. It is noted that Poison type III and Treatment A have the fastest mean of rates.
Figure 3. Interaction Plots
In order to see whether these e ects are signi cant or not， we conduct a further two-way analysis of variance with interactions for 48 animals' rates to death against Poison type and Treatment. A two-way analysis of variance with interactions gives the following output in R （Table 2）：
Response： rates（per hour）
Table 2. ANOVA Table The p-value of Interaction（0.387） is larger than 1%， thus we cannot re-ject the null hypothesis and get same statistical inference as that we gained in statistical analysis of survival times： there is no evidence of interactions between Poison type and Treatment. Since the p-values of Poison type and Treatment are far less than 0.001. There is a strong evidence that the main e ects of two factors in uence the rates of animals to death. Fur-thermore， the normal plot and plot of residuals below（Figure 4） show that both assumptions appear to be satis ed： rates of 48 samples of animals are normally distributed with constant variance. This statistical analysis based on Normal linear model for rates is more reasonable than previous one.
Figure 4. Diagnostic Plots
3Two statistical analyses comparison
From Plots of residuals and Normal plots in above section 2， we can see that the second analysis of rates improves the Normal linear model for this two-way classi cation compared with the rst analysis of survival times. We
can also compare conclusions of two analyses by using group mean table. Table 3 and 4 are group mean tables of survival times and rates respectively：
Table 3. Group Mean Table
We note that the survival times are generally higher in Poison type I and Treatment B and lower in type III and Treatment A； the rates， through， which are reciprocal values of the survival times of 48 sampled animals， are generally larger in Poison type III and Treatment A while smaller in type I and Treatment B.
Table 4. Group Mean Table
Therefore， the conclusions based on Group Mean Table agree well with the results gained from interaction plots above. They all indicate that these two statistical analysis reach the same conclusion.
4Conclusion
In an experiment of studying the e ects of certain toxic agents， the in uence from two factors： Poison type and Treatment were examined. There are three types of poison and four di erent treatments given in this experiment. Each group of four animals were assigned randomly to a combination of one type of poison and one treatment. We are interested in the main e ects of two factors on survival times and rates to death of total 48 animals.
Two statistical analysis with respect to survival times and rates all show that there is no signi cant interactions between two factors. And the analyses reach the similar conclusions to this experiment： Poison type III and Treatment A have most e ective toxicity and fastest rates to death to 48 sampled animals. While Poison type I and Treatment B have most slightly toxicity and slowest rates to death to animals.
1Introduction
A designed experiment was conducted into the e ects of certain toxic agents on 48 samples of animals. The given dataset contains survival times of 48 animals. We are interested in examining the in uence of two factors： Poison type and Treatment on survival times and rates of 48 animals. There are three poison types（I， II， III） and four treatments（A， B， C， D）， a group of four animals were randomly allocated to each combination of two factors， which means four animal samples of each of the three poison types were sent to each treatment. In this report， we conduct two statistical analyses of survival times and rates respectively. We then compare two analyses and summarise a conclusion to this experiment in the last section.
2Statistical analysis
In this experiment， there are two qualitative explanatory variables： Poison type and Treatment. Since there is also the possibility that the e ect of changing the treatment depends on the poison type or， vice versa. This e ect between two classi cations is known as interaction e ect， therefore， we perform a two-way analysis of variance with consideration of interactions between two factors for statistical analyses.
2.1Statistical analysis of survival times
In the statistical analysis of survival times， there is one quantitative response variable： survival times of 48 sampled animals. Firstly， we are interested in whether interactions between Poison type and Treatment on survival times is signi cant or not.
Representing the data： we illustrate interactions between Poison type and Treatment in the interaction plots below （Figure 1）. As is shown in Figure 1， the lines on each plot are roughly parallel. But for some combina-tions of two factors， there are interaction e ects on survival times.
Figure 1. Interaction Plots
From Figure 1， we can observe that Poison type I and Treatment B have longest mean survival times， while Poison type III and Treatment A have the shortest. In order to test whether the overall interactions between e ects of Poison type and Treatment is signi cant or not， we need to conduct a further two-way analysis of variance with interactions for 48 animals' sur-vival times against Poison type and Treatment.
The two-way analysis with interactions gives the following output in R：
Response： Survival times（hours）
Table 1. ANOVA Table The fth column （Pr（>F）） in the above table contains p-value， which is the signi cant probability of each test statistic. The p-value of Interaction term（0.1123） is more than 1%， thus we retain our null hypothesis which means there is no evidence of overall interactions between the e ects of two factors in the interaction test. However， p-values of Poison type and Treatment are both far less than 0.001. Thus there is a strong evidence that main e ects of Poison type and Treatment in uence the survival times： average survival times are not same across three poison types and four treatment groups. The e ects of two factors on survival times are individually signi cant.
Besides， a normal plot of the residuals based on above two-way analysis of variance（right gure below） shows the assumption that the survival times are normally distributed is not quite satis ed since there exists many outliers
far away from the straight line in Normal Q-Q plot. And a plot of residuals against the tted values（left gure below） indicates that variance increases obviously with expected value. Thus the assumption of constant variance is also unsatis ed， and since both assumptions are invalid， this statistical analysis based on Normal linear model is not quite reasonable.
Figure 2. Diagnostic Plots
2.2Statistical analysis of rates
The designed experiment is also interested in analysing the rates of animals to death， which are de ned as reciprocal values of the survival times with
unit as hours1 ： Now we want to examine the e ects of di erences in Poison type and Treatment as well as interactions between two factors on the
rates to death of animals.
We illustrate interactions between Poison type and Treatment in the interaction plots below（Figure 3）. Since the lines on each plot are roughly parallel， there is little evidence of interactions between the e ects of two factors. But again for this analysis， both Poison type and Treatment have substantial e ects on the response. It is noted that Poison type III and Treatment A have the fastest mean of rates.
Figure 3. Interaction Plots
In order to see whether these e ects are signi cant or not， we conduct a further two-way analysis of variance with interactions for 48 animals' rates to death against Poison type and Treatment. A two-way analysis of variance with interactions gives the following output in R （Table 2）：
Response： rates（per hour）
Table 2. ANOVA Table The p-value of Interaction（0.387） is larger than 1%， thus we cannot re-ject the null hypothesis and get same statistical inference as that we gained in statistical analysis of survival times： there is no evidence of interactions between Poison type and Treatment. Since the p-values of Poison type and Treatment are far less than 0.001. There is a strong evidence that the main e ects of two factors in uence the rates of animals to death. Fur-thermore， the normal plot and plot of residuals below（Figure 4） show that both assumptions appear to be satis ed： rates of 48 samples of animals are normally distributed with constant variance. This statistical analysis based on Normal linear model for rates is more reasonable than previous one.
Figure 4. Diagnostic Plots
3Two statistical analyses comparison
From Plots of residuals and Normal plots in above section 2， we can see that the second analysis of rates improves the Normal linear model for this two-way classi cation compared with the rst analysis of survival times. We
can also compare conclusions of two analyses by using group mean table. Table 3 and 4 are group mean tables of survival times and rates respectively：
Table 3. Group Mean Table
We note that the survival times are generally higher in Poison type I and Treatment B and lower in type III and Treatment A； the rates， through， which are reciprocal values of the survival times of 48 sampled animals， are generally larger in Poison type III and Treatment A while smaller in type I and Treatment B.
Table 4. Group Mean Table
Therefore， the conclusions based on Group Mean Table agree well with the results gained from interaction plots above. They all indicate that these two statistical analysis reach the same conclusion.
4Conclusion
In an experiment of studying the e ects of certain toxic agents， the in uence from two factors： Poison type and Treatment were examined. There are three types of poison and four di erent treatments given in this experiment. Each group of four animals were assigned randomly to a combination of one type of poison and one treatment. We are interested in the main e ects of two factors on survival times and rates to death of total 48 animals.
Two statistical analysis with respect to survival times and rates all show that there is no signi cant interactions between two factors. And the analyses reach the similar conclusions to this experiment： Poison type III and Treatment A have most e ective toxicity and fastest rates to death to 48 sampled animals. While Poison type I and Treatment B have most slightly toxicity and slowest rates to death to animals.