Team G4: Ying Zhang, Yiming Jin, Ziyue Gu Professor Leonard
ISA 365A
12/09/2018
The Candle Experiment –Final project Part 1: Full factorial design
Introduction:
For this project, we aimed to test the factors that affect candle burning. First, we used a 23Full factorial design in our initial experiment, which has three factors. The candles we selected are the common birthday candle (see picture 1) which are sold at Kroger. We used this kind of candle to measure the time it takes to burn. The three factors that we selected are the angle at which the candle was, whether the candle was frozen and whether the candle was in a can. For each factor, there are two levels. Firstly, for the variable angle, we took the horizontal surface as the reference, we put the candle sideways, which is parallel to the horizontal plane, which was equal to the zero. In addition, the candle is placed vertically at a ninety-degree angle. For the frozen factor, it was a categorical variable which has two levels, frozen and unfrozen. Also, for the factor of the can which is also the categorical variable with two levels, shield and without the shield.
Response variable: Burning time (in minutes)
Predictor variables: Angle (90 degree(vertical), 0 degree(horizontal));
Frozen (without frozen, with frozen);
Can (without a shield, with shield). Explanatory of level: Angle (L1 L2); Frozen (L1 L2); Can (L1 L2)
Picture 1: candle
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Data collection:
We began the experiment by setting up the data table with predictors. For the 23Full factorial design, we selected angle of the candle, whether it was frozen, and whether it was canned as the factors. For each factor, there were two levels. We decided to do two replicates because it would be more accurate rather than a single replicate. In addition, we did sixteen runs for the 23 full factorial design. With this, the total degree of freedom is 15. By making the table (see Table 1), we had a total of seven columns, this includes pattern, angle, frozen, canned, burning time, run order, and residual. For the run order, we used the random number to avoid systematic error and the residual which we will use it to check the assumption for the model. For the factor “angle,” we denoted horizontal as 0 degrees and vertical as 90 degrees. The candle that we used for the experiment has the bottom which is convenient for us to hold the candle to change the angle of the candle. Furthermore, for the factor of “frozen,” we put half of the sixteen candles into the fridge at the temperature of 28 for one night, and the other half keep the normal temperature. We record the time
Table 1
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°F
when the candle burns to the level of the chassis.
Analysis:
First, we fit the model with all the effects, two-factor interactions, and three-factor interaction. Here, we have three main effects, three two-factor interactions, and one three-factor interaction. We estimate that the fastest time for a burning candle is horizontal(0 degree), frozen and with a can. While the longest time for a burning candle is vertical(90 degrees), unfrozen and without the can. We assumed the fitted model with all the effects and interactions is:
𝑦̂=𝛽 +𝛽𝑥 +𝛽𝑥 +𝛽𝑥 +𝛽𝑥𝑥 +𝛽𝑥𝑥 +𝛽𝑥𝑥 +𝛽𝑥𝑥𝑥 01122334125136237123
Figure 1
The first thing we did to analyze our data is to create a boxplot. Figure 1 shows that the boxplot compares “can” and “angle” overlaid with “frozen” in relation to the burning time. Referring to the can, the boxplot shows that there is a decrease in burning time from candles without can to the candle with the can. Additionally, there is an increase in burning time for “angle” as it goes from horizontal to vertical. In conclusion, the candle without a shield burns a longer time, in addition, a higher angle(vertical) leads to a longer burning time. As we seen in this graph, the burning time of frozen candles at zero degree of angle between with can and without a can is significantly different. There is still one thing very questionable that we suppose that the candle without frozen burns longer than the frozen candle but looking back our boxplot we only saw
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that candle burning without can at two different angles shows our expected results. In another group of the experiment without cans, the burning time of candle unfrozen at angle 0 is longer than a frozen candle, which has a conflict with our hypothesis. There seemed to be no systematic error in our experiment because the points are super constant. Looking back at our conversation with Professor Leonard, we determined that we did make some mistakes in the experiment. We want to explain one by one in our conclusion of the initial experiment.
Figure 2
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By looking at figure 2, we can compare the p-value to remove the insignificant terms. As we did the two replicates, the degree freedom of total is 15 the degree freedom of model is 7, and the degree freedom of error is 8. R-squared is equal to 0.994857, and the RMSE is 0.319316. We supposed the significance level was 0.01. As the p-value of the Frozen*Can is equal to 0.01860 which is greater than the significance level, so we decided to remove this point. Other points were pretty well placed, and we kept them because of the small value of the p-value.
After Removing:
Figure 3
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After we removed the Frozen*Can, from figure 3, we left the Angle, Frozen, Can, Angle*Frozen, Angle*Can and Frozen*Can*Angle which have the p-value less than the significance level 0.01. In addition, we found that there is a little change of the R-square value, which dropped from 0.994587 to 0.988723. We could say that the interaction with Frozen*Can is insignificant. RMSE is equal to 0.434524. As the SST and SSE were changed from 149.87130 to 148.98770, and from 0.81570 to 1.69930 respectively. Because the SST dropped and the SSE increased which means the data is less far from the mean and the data is farther from the predicted values. Besides, the F-ratio is changed from 209.9810 to 131.5139. It determines the proposed relationship between the burning time and Angle, Frozen, Can, Angle*Frozen, Angle*Can, and Frozen*Can*Angle. As seen in the ANOVA table, the degree freedom of total is 15, the degree freedom of model is 6, with Angle, Frozen, Can, Angle*Frozen, Angle*Can, and Frozen*Can*Angle. The degree freedom of error is 9. Finally, we get the final model from the summary table:
𝑦̂ = 7.685 − 1.105𝑥𝐴𝑛𝑔𝑙𝑒 + 0.66375𝑥𝐹𝑟𝑜𝑧𝑒𝑛 + 1.28875𝑥𝐶𝑎𝑛 + 1.59125𝑥𝐴𝑛𝑔𝑙𝑒𝑥𝐹𝑟𝑜𝑧𝑒𝑛 + 0.70625𝑥𝐴𝑛𝑔𝑙𝑒𝑥𝐶𝑎𝑛 +
1.72𝑥𝐹𝑟𝑜𝑧𝑒𝑛𝑥𝐶𝑎𝑛𝑥𝐴𝑛𝑔𝑙𝑒
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Figure 4
Figure 4 illustrates the interaction plot of Angle, Frozen, and Can respectively. As the plot shows the interaction of Frozen and Can, the lines are parallel which means there is no interaction between “Frozen” and “Can.” So, that is the reason why we removed this interaction effect in the initial model before. Furthermore, by plotting the 0 degree and 90 degrees effects of Angle with Frozen at not frozen and with frozen across the x-axis, the lines have different slopes. So, it has interaction. The same way, two lines across the Angle and Can have different slopes, which indicates it has an interaction. Thus, we kept these two interactions, Angle*Can, and Angle*Frozen in our model.
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Figure 5
Figure 5 shows the histogram and Normal QQ-plot of residuals. Most of the effects were close to the center(zero) line seemed that follow the fitted normal model straight line. The normal quantile plot of the residuals and histogram shows a fairly symmetric, unimodal distribution.
Figure 6
By looking at the residuals by the predicted plot (figure 6), it seems that has the visible pattern in this residual plot. Since the range of the residual spread between -0.5 and 1, there is a decreasing trend from the left side to right side. So, one can say that residuals have non- constant variance.
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Figure 7
Figure 7 shows the residuals in respect to the run order. This plot demonstrates that the errors are independent of one another as the residuals appear to be randomly scarred with no pattern to them. There are no outliers that are far from the predicted values. Thus, we assume that the error is independent of each other.
Conclusion:
In conclusion, we found that all three factors have a significant influence on burning time since all p values in figure 2 looked good. We decided to use 0.01 as our significant level to evaluate if any terms model should be removed before arriving at our final model. We promoted our first initial model by removing the interaction effect of “frozen” and “can.” We rejected our hypothesis that the candle burning at 90 degrees of angle unfrozen and with a shield had the greatest burning time. As seen in the data table (table 1), the configuration of 0 degrees of angle, not frozen and without can be caused the candle to burn the longest time for both replicates. Run 5 had a burning time of 12.45 minutes and run 13 had burning time 12.18 minutes. Additionally, the figuration of 90 degrees of angle, with frozen and without can also be caused the candle to burn near equivalent time. Run 3 had a burning time of 12.26 minutes and run 11 had a burning time of 12.25 minutes. Therefore, our hypothesis holds false regarding which levels resulted in the longest burning time.
As we mentioned before, we predicted the effect of both two levels of each factor. In our analysis for the boxplot (figure 1), we concluded that the “angle” factor with higher level (90 degrees) and the “could” factor with lower level (without) lead to longer burning time. This could be partially a result from frozen factor being found rejecting to our hypothesis, given that we made some mistakes in the process of our initial experiment. We took out all frozen candles from the fridge at the same time when we got started to do an experiment, which may result in different temperatures of each candle. We also ignored the time interval between taking out the candles and doing each run of our experiment. Looking back on our conversation with Professor
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Leonard, we realized that we made a design mistake. We did not choose an appropriate size for can so that the candles did not fit the shield well. In this case, there would not be enough space for burning, which may cause candles to burn in less time than we expected — another problem that could be consideration of choosing factor levels. We selected too extreme levels of angle factor. If we were to redo the experiment, we would choose two levels very close to each other.
Based on our initial experiment, the first thing we needed to do is keeping the frozen candles at the same temperatures. We decided to take only one candle for each run instead of taking all of them out from the fridge at the beginning of our experiment. We also would like to focus more
on how much the “frozen” factor influenced on burning time so that we may determine how to design an experiment by changing frozen factor only. Since from boxplot we saw that burning time of frozen candles at zero degree of angle between with can and without can were significantly different, we would concentrate on effects of sizes of the can in our next experiment.
Part 2: CRD design and Full factorial design
Introduction:
Experiment 1: The purpose of the design of this experiment is to investigate the influence of frozen factor on the burning time of candles. We supposed angle at forty-five degrees and kept using no shield this time. As we did before, we put the candle into the fridge at a temperature of 28°F for one night. There will be two levels for frozen factor, “frozen” and “unfrozen” After discussing with professor we determined to do four replicates in this experiment. We suspected that candles which were not frozen would burn longer time than frozen ones did.
Experiment 2: The purpose of the design of this experiment is to investigate the influence of the “can” factor and “frozen” factor on the burning time. Compared to our initial experiment, we selected two levels by changing the size of the can. We supposed the angle at ninety degrees in this experiment. As we did before, we put the candle into the fridge at the temperature of 28.°F for one night. We suspected the configuration of the big can and without frozen would cause candles to burn the longest time. For testing, we were permitted to perform 2X2 factorial design, where we chose 2-factor levels to test each influence. There would be four replicants in this experiment.
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Experiment 1 Data collection:
For table 2, we had four columns with Frozen, Burning time(minutes), Run order, and residual. This is a single factor experiment. We did four replicates for this experiment because replication reduces variability in experimental runs. Based on the previous experiment, we originally recorded the burning time when the
By the way, in order to avoid the systematic error, we choose to take the candle one by one from
the fridge to do the experiment.
candle burns to the level of the chassis. But we noticed that
there is an error in the degree of burning of the candles observed by each of us. So, we decided
to record the time until the whole candle is all burned, here we mean when the fire goes out.
Table 2
Firstly, we did a completely randomized design with a single factor. As the box-plot(figure 1) shows that the result of the burning time without frozen at angle 0 is longer than frozen candles has a conflict with our hypothesis. We hypothesized that the forty-five-degree angle and candles without a can would change the frozen factor.
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Analysis:
Figure 8
Figure 8 shows the boxplot comparison for each of two levels; the candles burn time from frozen and unfrozen. From the boxplot, we saw that the burning times were significantly different between two levels for four replicants. The candles which were not frozen burned longer than frozen ones did. These results are as same as our original prediction because we inferred that the unfrozen candles would burn much longer time.
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Figure 9
From Figure 9, the summary of the second experiment shows that the p-value of frozen/no frozen looks good individually. As the p-value is less than the significance level(𝛼 = 0.01), we could say frozen is the significant factor. The F Ratio is 198.4648 and the R-square is 0.970655,
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which indicates that 97.07% variation of burning time can be explained by frozen/no frozen factor. There were four replicates in our one-factor experiment with 8 runs total which give us 7- degree freedom of total. The degree freedom of model is one while the degree freedom of error is six. The mean burning time for the candles without frozen is around 2 minutes longer than the candles which were frozen.
Figure 10
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The least square mean for the unfrozen level was 12.65 minutes, with frozen the mean was 10.53 minutes. It does appear to support our conclusion from boxplot because it depicted that candles burn without frozen had longer burning time.
Conclusion
In this experiment, we choose forty-five degrees between zero degree and ninety degrees as the angle. This would allow us to hone in the frozen factor that impacted on burning time most. The p-value in our ANOVA table is less than 0.0001, which indicated that we failed to reject our hypothesis that frozen factor has no impact on burning time. Looking back on our experiment, we found that our original prediction of initial 23Factorial design was true, which is that candles that were unfrozen burned a longer time than frozen ones did. This demonstrated that our guess and reasons for making mistakes in the process of initial 23Factorial experiment holds true. Therefore, we realized that how it was important to control other factors that might have an impact on our experiment. Lastly, we were unable to identify any issues of confounding and systematic error.
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Experiment 2 Data collection:
Table 3 has six columns, pattern, can, frozen, burning time, run order, and residual. In this experiment, it has two factors, Can and frozen. For the Can, there are two level, small can, and big can. The can that we used in the experiment are the common can that we could see in the Kroger, small can has the 12ozl volume while big can has 16 oz volume. In addition, for the factor frozen, it has two levels, frozen and unfrozen. We put the candle into the fridge at a temperature of 28°F for one night, and others are not placed in the refrigerator to maintain a normal temperature. We hypothesized the angle keeps the same with vertical (90 degrees). As the experiment is the 22Design, it at least has four runs. We decided to do four replicates, and the degree freedom of total is 15, the degree freedom of model is 3, and the degree freedom of error is 12.
Response variable: Burning time(minutes) Predictor variables: Can(small can, big can);
Frozen(no frozen, frozen).
Table 3
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Figure 12
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The first thing we did to analyze our data was to create a boxplot. The boxplot we created, compared the frozen factor and can factor to the mean burning time. Referring to frozen factor, the boxplot shows that there is a decrease in burning time from low(with frozen) to high(without frozen). In addition, there is an increase in burning time for can factor when it goes low(small size) to high(big size). In conclusion, having a lower frozen caused the candles to burn longer, and having a higher can size also led to a longer burning time.
Analysis:
We fit the full model with all main effects Can and Frozen and 2-factor interaction, Can*Frozen. Therefore, we get the fitted equation as follow:
𝑦̂ = 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 + 𝛽 3 𝑥 1 𝑥 2
Figure 13
Figure 13 estimates the summary table for the initial model that we fitted before. The R-squared for the initial model is equal to 0.96397, and the RMSE is equal to 0.325877. We could say that around 96.397 of the variation in burning time can be explained by the linear predictors Can and Frozen. As the p-value of the can and frozen are both less than the significance level(𝛼 = 0.01), we could say that these two effects are significant. The p-value of Frozen*Can is 0.9401 which is greater than the significance level, we removed it as an insignificant term.
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After removing:
Figure 14
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After removing the insignificant term, Frozen*Can, we found that the p-value of Can and frozen both less than the significant level(𝛼 = 0.01), we can say they are the significant factor. There is a little change of the R-square. It dropped from 0.96397 to 0.963952. And the RMSE changed from 0.325877 to 0.313169. Thus, we get the newly fitted equation:
𝑦̂ = 8.0825 − 1.40875𝑥𝐶𝑎𝑛 + 0.3825𝑥𝐹𝑟𝑜𝑧𝑒𝑛
Finally, we found that as we did the 23full factorial experiment before, we removed the Frozen*Can term because of the large p-value and insignificant. In this experiment, we also removed the Frozen*Can, which indicates that there is no relation between Frozen*Can and burning time.
Figure 15
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The first assumption that we are able to check out is that the errors are normally distributed. We know this by looking at histogram and QQ plot of the residuals for the final model in figure 8. The histogram does appear to be unimodal but is not fairly symmetric. It seemed to skew to the right. In regards to the QQ plot, the residuals appear to follow the reference line closely.
Figure 16
Figure 16 shows the plot of residuals versus the predicted response; the separation was above and below zero. And the range of the points are between -0.6 and 0.4. We could say that the residuals have constant variance.
Figure 17
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By looking at the residual burning time (in minutes) vs. the run order(figure 17), we could see that the separation of the points is around the zero at the range of 4. Besides, there is no visible pattern in this plot; we assume the residuals have constant variance.
Figure 18
The prediction profiler in figure 18 describes the relationship between the small can, big can, frozen, no frozen and burning time. As we estimated that small can would spend less time than big can and factor Can have the relation with burning time. Thus, it is the reason why we removed the interaction between frozen and Can.
Conclusion
In conclusion, we found that the greatest influence was the interaction effects of a frozen factor and can factor. We did predict the relationship between the frozen factor and can factor after discussing with Professor Leonard. In this experiment, we selected two levels by changing sizes of the can. Compared to our 23Factorial experiment, we changed those two types of cans into bigger sizes so that there is enough space for candles to burn. As seen in the data table (table 3), the configuration of the unfrozen candles and big can size caused candles to burn the longest time. Run 7 had a burning time of 10.12 minutes, and run 8 had a burning time of 10.23 minutes. Therefore, our hypothesis holds in regards to which level resulted in the longest burning time. Based on our final model, our educated guess would be that in bigger size of the can, there would be more space for letting oxygen into the can, which caused candles to burn longer.
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Part 3: Experiment design
Response Variable: Burning time (in minutes)
Predictor variables: Angle (zero degree, forty-five degree, ninety degrees);
Frozen (without frozen, with frozen); Can (without a shield, with shield).
In our previous experiments, “angle” and “can” were the predictors which could be continuous. We selected angle and decided to change its variable type be continuous in our third experiment because we believe it is a very interest topic. From Table 1, the unfrozen candle which placed inside the can had the longest burning time in the horizontal direction while the candle with frozen had the longest burning time in the vertical direction. We can conclude that the candles without the can be burned a longer time compared with the candles which were placed inside the can. However, there are several strange data points. Thus, it is hard to conclude that the higher angle or lower angle causes the longer burning time. From Table 2, the data shows that the average burning time for unfrozen candles is around 2 minutes longer than the frozen candles. However, this conclusion did not show in the first experiment. It is possible for angle and frozen/no frozen to have correlated relationship. We want to see if there is a different result when the angle is the 45-degree angle which is between the vertical direction and horizontal direction. In this experiment, there are three levels for “angle”, two levels for “frozen” and 2 levels for “can” which allow us to have 12 runs. We decided to do two replicates because it would be more accurate rather than a single replicate. The time for the candles burns to the bottom will be the response variable to check the result of the experiment. We have already had all the measurement we need for each run from the initial experiment and the second part. We did lots of experiment in the initial one and the second part. We combined the second experiment with the initial dataset to come up with the new experiment. Although we have already had the measurement for all the runs in this experiment, we still want a hypothesis for the designed experiment. Our hypothesis for the third experiment is that a candle without frozen and can which burned in the vertical direction will burn for the longest time. Also, it is important for us to investigate the possible quadratic of the angle variable. The environment for this experiment should be within the room to avoid the error caused by wind and other environmental factors.
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Table 4
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