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- What type of MANOVA would be used with more than one dependent variable and one independent variable with more than two levels? What type of MANOVA would be used with more than one dependent variable and more than one independent variable which all...
- It answers the question Is the mean of at least one group different than the mean of other multiple groups of data? It's likely to be one of the most common test you will use as a Six Sigma project manager. Assumptions Each sample is normally...
- WITHIN sample variance explains the variation within each sample itself look at a Box Plot of one data set to graphically comprehend this - the tip of one whisker to another. ANOVA answers the question if the means of several populations are statistically different or equal. The t-test are limited to comparing up to just two groups. Using ANOVA to compare two sample means is equivalent to using a t-test to compare the means of independent samples. Inference Space: Range of the factors being evaluated.
- Fit: Predicted value of the POV y with a specified setting of factors. Residual: Difference from the fit and actual experimental output. Removing the one sample could completely change the result of the test. That is why visual depiction, such as Box Plots, can help find the drivers to the test result or samples that are flawed. If the Null Hypothesis, Ho, is found to be true, then we would not expect to see a lot of variation Between Samples. All the population means are considered equal. If Ho is not true, expect to see significant variation between the samples. This would imply that the difference between samples is large relative to the variation within samples. Reminder: Statistical significance does not always imply practical significance. Every numerical result needs to taken under scrutiny to determine if it makes sense in reality. Looking at the Box Plot and Confidence Intervals are easy way to pick them out.
- Fisher's Pair-Wise comparison is another statistical method. Calculate Epsilon-squared. A low value may indicate that other factors may exist. Review statistical conclusion and state the practical conclusion. State the level s that are different if such is determined. The sample sizes do not have to be equal. Determine if there is a significant difference of means in two or more appraisers. The results of a mock study where four appraisers were timed to make an inspection decision on a 13 widgets.
- All other criteria are equal. There are four levels that are controlled in the experiment, one being each appraiser. The first step is to create the test. In general, if the p-value is lower than the alpha-risk then the alternate hypothesis is inferred reject the null. Hypothesis Test: Null Hypothesis: Population means of the different appraisers are equal. Using a One-Way test with an alpha-risk of 0. The F -statistic , and heavily overlapping confidence intervals are also evidence that there is no difference among any pairs or combinations of them. It is concluded that there is not a statistical difference between any of the appraisers. What if? It doesn't conclude which one Other notes: Paul has the lowest average time per appraisal but Jim has lowest variation and the most consistent time for each appraisal. With these results a Six Sigma Project Manager would likely be very pleased that all are performing the same in terms of time spent making an appraisal and the variation from appraisal to appraisal is similar among each person hopefully the correct appraisal too.
- This is likely a result of consistent training and adherence to the SOP's. However, the next questions from the Six Sigma Project Manager is Caution: It still may be possible that seconds per appraisal is not acceptable by the company, or customer, and this still needs to be reduced. This test is not comparing the appraisers to a target value.
- The F-critical value is 2. You can use the F-table above to get a close estimate of the F-critical value. One downfall with tables is sometimes you may not get a precise number since not every combination is shown. However, the table can provide a fairly good estimate and at least allow a decision to be very conclusive. The numerator has 3 degrees of freedom and the denominator has 48 degrees of freedom. Using the table below shows that the F-critical value is going to be between 2. And in this case, both values are much higher than the F-calculated value of 0. As a Six Sigma project manager it may be worth re-running depending on cost and time the trial with a larger sample size and additional appraiser training to reduce the variation within each one.
- The variation is fairly consistent among each of them so it appears there is a systemic issue that is causing nearly similar amounts of variation within each appraiser. It is possible that one or a few of the widgets are creating the similar spread in the timing for each appraiser. You may examine the timing performance of each widget and run an ANOVA among the 13 widgets and see if one or more stands out.
- This is 4. This is a low value so it is possible that other Factors exist that are creating the variation. Select ADD and the menu will pop up as shown on the right of the picture below. As you can see, there are several statistical tool to choose from. The team recorded the pieces per minute that were produced of the same PN XYZ under similar operating conditions and had to be acceptable pieces. They wanted to examine several things with one of them being if any of the machines mean performance varied from the other. Recall, that sample sizes do not have to be the same. Understanding the basic meaning and applications for this commonly used test is necessary for any level of a Six Sigma Project Manager. Factors are differences in things such as, but not limited to, parts produced its probably not a good idea to compare the production of pencils to the production of nails even if they run on similar machines , services delivered, time, different operating conditions, and customer requirements.
- Before jumping into a multivariate analysis, use ANOVA to focus on one factor at a time and learn from that analysis first, then use multivariate if something significant is found. Once the data is collected the ANOVA takes very little time and evaluating the factors in various ways only provides more and more insight as to their relationship.
- What is your computed answer? What would be the null hypothesis in this study? What would be the alternate hypothesis? What probability level did you choose and why? What were your degrees of freedom? Is there a significant difference between the four testing conditions? Interpret your answer. Explain your answer. Answer Neuroscience researchers examined the impact of environment on rat development. Rats were randomly assigned to be raised in one of the four following test conditions: Impoverished wire mesh cage - housed alone , standard cage with other rats , enriched cage with other rats and toys , super enriched cage with rats and toys changes on a periodic basis.
- After two months, the rats were tested on a variety of learning measures including the number of trials to learn a maze to a three perfect trial criteria , and several neurological measure overall cortical weight, degree of dendritic branching, etc. The data for the maze task is below. Compute the appropriate test for the data provided below.
- The F ratio is a convenient measure that we can use to test the null hypothesis. This indicates that the independent variable did not affect the dependent variable, so we cannot reject the null hypothesis. This indicates that the independent variable did affect the dependent variable, so we must reject the null hypothesis. What does it mean for the F ratio to be significantly greater than one? To answer that question, we need to talk about the P-value. P-Value In an experiment, a P-value is the probability of obtaining a result more extreme than the observed experimental outcome, assuming the null hypothesis is true. With analysis of variance, the F ratio is the observed experimental outcome that we are interested in. So, the P-value would be the probability that an F statistic would be more extreme i.
Clinical Pharmacology And Pharmacokinetics: Questions And Answers
We can use Stat Trek's F Distribution Calculator to find the probability that an F statistic will be bigger than the actual F ratio observed in the experiment. Enter the between-groups degrees of freedom 2 , the within-groups degrees of freedom 12 , and the observed F ratio 4. Therefore, the P-Value is 0. Hypothesis Test Recall that we specified a significance level 0. Once you know the significance level and the P-value, the hypothesis test is routine. Here's the decision rule for accepting or rejecting the null hypothesis: If the P-value is bigger than the significance level, accept the null hypothesis.- If the P-value is equal to or smaller than the significance level, reject the null hypothesis. Since the P-value 0. And we conclude that the mean cholesterol level in at least one treatment group differed significantly from the mean cholesterol level in another group. Magnitude of Effect The hypothesis test tells us whether the independent variable in our experiment has a statistically significant effect on the dependent variable, but it does not address the magnitude of the effect. Here's the issue: When the sample size is large, you may find that even small differences in treatment means are statistically significant.
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When the sample size is small, you may find that even big differences in treatment means are not statistically significant. With this in mind, it is customary to supplement analysis of variance with an appropriate measure of effect size. Eta squared is the proportion of variance in the dependent variable that is explained by a treatment effect. It appears that the relationship between dosage level and cholesterol level is significant not only in a statistical sense; it is significant in a practical sense as well. The analysis that we just conducted provides all of the information that we need to produce the following ANOVA summary table: Analysis of Variance Table.- Unfortunately, we have to warn you that you might find this next stuff a bit complicated. You might not, and that would be great! We will try our best to present the issues in a few different ways, so you have a few different tools to help you understand the issue. For example, up until now we have been talking about experiments. Most every experiment has had two important bits, the independent variable the manipulation , and the dependent variable what we measure. In most cases, our independent variable has had two levels, or three or four; but, there has only been one independent variable. What if you wanted to manipulate more than one independent variable? If you did that you would at least two independent variables, each with their own levels.
- The rest of the book is about designs with more than one independent variable, and the statistical tests we use to analyze those designs. We will be imagining experiments that are trying to improve students grades. So, the dependent variable will always be grade on a test. Time of day Morning versus Afternoon : Do students do better on tests when they take them in the morning versus the afternoon? There is one IV time of day , with two levels Morning vs. Afternoon Caffeine some caffeine vs no caffeine : Do students do better on tests when they drink caffeine versus not drinking caffeine?
- There is one IV caffeine , with two levels some caffeine vs no caffeine 1 IV three levels : We would use an ANOVA for these designs because they have more than two levels Time of day Morning, Afternoon, Night : Do students do better on tests when they take them in the morning, the afternoon, or at night? There is one IV time of day , with three levels Morning, Afternoon, and Night Caffeine 1 coffee, 2 coffees, 3 coffees : Do students do better on tests when they drink 1 coffee, 2 coffees, or three coffees?
- Afternoon ; IV2 Caffeine: some caffeine vs. We had students take tests in the morning or in the afternoon, with or without caffeine. IV1 Time of day has two levels morning vs afternoon. IV2 caffeine has two levels some caffeine vs. The first two designs both had one IV. The third design shows an example of a design with 2 IVs time of day and caffeine , each with two levels. This is called a 2x2 Factorial Design. It is called a factorial design, because the levels of each independent variable are fully crossed. This means that first each level of one IV, the levels of the other IV are also manipulated. We apologize for that. We said this means the IVs are crossed. To illustrate this, take a look at the following tables. We show an abstract version and a concrete version using time of day and caffeine as the two IVs, each with two levels in the design: Figure 9. For the first level of Time of Day morning , we measure test performance when some people drank caffeine and some did not.
- So, in the morning we manipulate whether or not caffeine is taken. Also, in the second level of the Time of Day afternoon , we also manipulate caffeine. We could say the same thing, but talk from the point of view of the second IV. For example, when people drink caffeine, we test those people in the morning, and in the afternoon. So, time of day is manipulated for the people who drank caffeine. Also, when people do not drink caffeine, we test those people in the morning, and in the afternoon, So, time of day is manipulated for the people who did not drink caffeine. Finally, each of the four squares representing a DV, is called a condition. So, we have 2 IVs, each with 2 levels, for a total of 4 conditions. This is why we call it a 2x2 design. The notation tells us how to calculate the total number of conditions.
- We use a notation system to refer to these designs. The rules for notation are as follows. The number of levels in the IV is the number we use for the IV. The first IV has 2 levels. The second IV has 3 levels. The third IV has 2 levels. There are a total of 12 condition. Figure 9. As you can see there are now 6 cells to measure the DV. This immediately makes things more complicated, because as you will see, there are many more details to keep track of. Why would researchers want to make things more complicated? Why would they want to manipulate more than one IV at a time. When you have one IV in your design, by definition, you are manipulating only one thing. This might seem confusing at first, because the IV has more than one level, so it seems to have more than one manipulation. Consider manipulating the number of coffees that people drink before they do a test. We could have one IV coffee , with three levels 1, 2, or 3 coffees. You might want to say we have three manipulations here, drinking 1, 2, or 3 coffees.
- But, the way we define manipulation is terms of the IV. There is only one coffee IV. It does have three levels. Nevertheless, we say you are only doing one coffee manipulation. The only thing you are manipulating is the amount of coffee. To do another, second manipulation, you need to additionally manipulate something that is not coffee like time of day in our previous example. Returning to our question: why would researchers want to manipulate more than one thing in their experiment. The answer might be kind of obvious. They want to know if more than one thing causes change in the thing they are measuring!
- If you wanted to track down how two things caused changes in happiness, then you might want to have two manipulations of two different IVs. This is not a wrong way to think about the reasons why researchers use factorial designs. They are often interested in questions like this. However, we think this is an unhelpful way to first learn about factorial designs. Effects are the change in a measure caused by a manipulation. You get an effect, any time one IV causes a change in a DV. Here is an example. We will stick with this one example for a while, so pay attention… In fact, the example is about paying attention. You could something like this: Pick a task for people to do that you can measure. For example, you can measure how well they perform the task. That will be the dependent measure Pick a manipulation that you think will cause differences in paying attention.
- For example, we know that people can get distracted easily when there are distracting things around. You could have two levels for your manipulation: No distraction versus distraction. Measure performance in the task under the two conditions If your distraction manipulation changes how people perform the task, you may have successfully manipulated how well people can pay attention in your task. First, we pick a task. You may have played this game before. You look at two pictures side-by-side, and then you locate as many differences as you can find.
- If you pay attention to the clock tower, you will see that the hands on the clock are different. One difference spotted. We could give people 30 seconds to find as many differences as they can. Then we give them another set of pictures and do it again. Every time we will measure how many differences they can spot. So, our measure of performance, our dependent variable, could be the mean number of differences spotted. If people need to pay attention to spot differences, then presumably if we made it difficult to pay attention, people would spot less differences. What is a good way to distract people? How about we do the following: No distraction condition: Here people do the task with no added distractions. They sit in front of a computer, in a quiet, distraction-free room, and find as many differences as they can for each pair of pictures Distraction condition: Here we blast super loud ambulance sounds and fire alarms and heavy metal music while people attempt to spot differences.
E: F Distribution And One-Way ANOVA (Exercises) - Statistics LibreTexts
We also randomly turn the sounds on and off, and make them super-duper annoying and distracting. But, we want to make them loud enough to be super distracting. We should find a difference! If our manipulation works, then we should find that people find more differences when they are not distracted, and less differences when they are distracted. For example, the data might look something like this: Figure 9. People found 5 differences on average when they were distracted, and 10 differences when they were not distracted.
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