BLOG ENTRY 2

Stacy Ribolini

August 10, 2014

Blog Entry #2---REPLIES

Make thoughtful, substantive comments to the 4 blog entries below. A,B,C,D

Your comments should be related to topics and issues discussed in the blog, and should further the mathematical thinking (by making connections, asking questions, answering questions posed, etc.) This is not a place for merely complimentary posts - though you may certainly include compliments as well.

A. I reflected in my last blog post that I have learned a lot about data collection and potential bias associated with carrying out a study. After completing the first stages of the comparative study project I was able to put what I had learned to good use. Over the past few weeks though I have definitely learned that creating the data collection tool is much easier than actually collecting the data itself. It is one thing to devise a plan for eliminating bias, but ensuring that the right conditions exist so that the biases don’t arise is extremely difficult. I found that as hard as I tried to think of every possible factor that could present a bias, there were still things that I had never even considered until I was in the store ready to collect the data. This process makes me wonder if it is really possible to have a study that is completely void of bias. Despite this fact I have found that the work we did in week three and four with central tendencies and measures of spread was useful, as I feel comfortable using it to analyze the data I worked so hard to collect. I am eager to hear how others have faired with their projects!

I had similar sentiments during week four as I did with week three; I was familiar with many of the concepts of spread individually, how they were calculated and what they represented, but I was amazed at what these values together could tell you about the spread of a data set. The range was the most familiar measure for me and I was convinced that this was a good way to measure the variation in a data set. I quickly learned that although it may be the easiest to calculate and interpret, it isn’t actually the greatest way to evaluate the variation in a data set. In fact measures like the IQR, MAD and SD actually provide a much better picture for how varied a data set is.

I can remember working with hypothesis tests and correlation coefficients in my high school statistics class, but I quickly realized that these were concepts that I had spent time memorizing, rather than fully understanding. I will admit that at first I was clueless at how to carry out a hypothesis test, even though I had heard terms like z-score, p-value, null hypothesis before. I simply had no idea how they all worked together and what I would actually gain from carrying out the test. I will admit that I am still a little confused on the difference between a z-score and a t-score, but am going to continue to investigate so that I have a better understanding of the difference (if there is one?). Regardless of the fact that I was pretty clueless of how to do this at the beginning of the week, I was easily able to see the usefulness of the test when I worked with the data of the week relating to SAT scores.

I entered this course, with the feeling that most of it would be review, as I have taken a few previous statistics courses and enjoy the topic. My prior knowledge though only scratches the surface of what this domain of mathematics has to offer. In week one we learned of two types of variables quantitative and categorical. I feel as though the past six weeks has given me ample exposure and work with quantitative data, I am hoping that the last two weeks cover what can be done when working with categorical data. I am one who likes to be able to ‘quantify’ things so working with categorical data may be a bit frustrating, but I am hoping to learn of strategies and techniques that can be used to quantify relationships between categorical variables (I am thinking this will be the case just from the title of session seven!).

Best of luck to all during the remaining two weeks,

Stacy

B. These last three weeks have increasingly taken me out of my comfort zone. On the bright side, the challenging concepts are what I think about after the computer is turned off and the notebook is closed; that is when I try to make sense of what I do not fully comprehend. This material is interesting and I want to understand it better.

The Good: I enjoyed applying my new (or renewed) understanding of standard deviation to normal distributions and learning about the empirical rule. I always wondered how percentiles were determined for SAT scores. Our investigations in Module 4 shed light on my question. I’m guessing that normal distributions and standard deviations come in to play when statistics for MCAS (Massachusetts standardized test) scores are calculated and released for the school districts to analyze. That was carried further in Module 5 with the activity using z-scores to zero in on probabilities not found with the empirical rule.

I also am enjoying learning about association and relationships of bivariate data. Scatter plots and line of best fit are in the 8th grade common core standards. The method for finding line of best fit in 8th grade is to create a line (similar to the uncooked spaghetti exercise) in the center of the points, and try to get approximately the same number of points above the line as below the line. Points on the line are good, too. In the 8th grade, students only find the equation for the line of best fit when the line is already provided for them. I liked how quickly the line of best fit was graphed (and the equation was provided) using Excel.

The Bad: I totally misinterpreted the SAT Prep question. I had to go back and make sense out of why the SAT Prep scores were greater than the provided sample of SAT scores. In the same module, I found the hypothesis testing investigation to be a challenge, as well. I could follow the Shakespeare video and calculations, but I am not confident that I could come up with a null hypothesis and alternate hypothesis and the p-value on my own (not yet, anyway).

The Ugly: Finding the line of best fit and the least squares regression line for scatter plots. I spent a lot of time looking up how to come up with a least squares regression line. I’m afraid I’m making it more complicated than it has to be. I like the idea of finding the line with the lowest SSE and using that equation to make predictions.

It’s not all bad, of course. I focus on what I don’t understand because I want to be able to use this material. I think the weekly DoW activities, and more significantly, our research projects, are an excellent way to help us to understand and apply what we are learning. I would love to be able to come up with a way to incorporate this (research projects or data sets that can be used for more than one lesson) with different units at the middle school level. It has made me invest in wanting to understand the material so I can work with it, and I would hope I could pass that along to my students.

C. W6 Blog – Richard Cummins

For this blog, I will take a look at two important areas, my growth and the connections to the 7th grade curriculum. In doing this, I went back and reviewed the progression of this course. As I have never developed a statistical study before, clarifying definitions, goals, and steps of a study were valuable in determining how I was going to put my study together. These were areas where I had to develop an understanding where there had been none before. I tried to put myself in the place of my students and think about how they would feel and think when they learn something new.

While I was familiar with the different graphic representations of data, the review of creating box plots, histograms, bar graphs, and scatter plots, especially with technology, was very helpful. As we have introduced 1-to-1 technology in our school, I hope to take these tools and use them for my students to better understand the concepts behind the math instead of getting bogged down doing the calculations.

The concept of center should be firmly engrained in my students by seventh grade. They should be able to calculate mean, median, and range without much review. The key I will stress this year is an understanding of center. “What does mean represent?” as conceptual idea instead of a calculation. In addition, understanding and ability to calculate IQR is also part of the 7th grade curriculum. Through this course, I have a much deeper understanding of concepts of center.

Sixth grade students are introduced to the idea of MAD, in 7th grade they need to be able to calculate and analyze it. I taught a short unit on MAD last year, but did it without the understanding I have now. I have taken away several activities that I will be able to use directly in my classroom. This is the point in the course where future concepts are higher level than what my students need to know. It is also important for me to understand these concepts, so I know where these students are headed in high school classes and beyond.

In terms of connections, the relationship between statistics and probability is strong. Using probability analysis of samples can be helpful in predicting future results. We generally teach these two units together, so even in our curriculum they are connected. One of the more important connections I try to make with my students is between equations, graphs, and tables. When students can move efficiently between these representations, they demonstrate a higher level understanding of concepts.

This brings me to the theme I have seen as we complete this course. The over-riding theme is, “How can we compare data?” It started with basic computations of mean, median, range, and mode. I need my students to understand that these may not give us enough information to compare or predict future results. The goal is for them to begin to move from calculation to analysis, a higher order thinking. While my students won’t see it in 7th grade, additional ways to analyze data is available. I had never explored regression before, but as I did, I felt this is one of the more important ways to look at sets of data. The activity that bought this out best was the TV/Life expectancy data. The strength of the correlation did not come out until the correlation coefficient was calculated. The scatter plot showed a relationship, but it was much stronger than visually demonstrated.

The only concept that I am struggling with is the Least Squares Regression. I am still going back to rework that part of module 6 to see if I can clarify my understanding.

D

Christine Palmer

Since the beginning of this course and especially in the last three weeks, I have enjoyed working through the activities dealing with statistical concepts. Specifically, in week four, we delved into measuring spread by looking at the mean absolute deviation, the IQR, and the standard deviation. With the activities we worked through, it became very evident that measures of center (mean, median, and mode) do not give us a full picture of a data set. In order to see what is going on throughout the data, it is very helpful to look at the IQR (if there are outliers present in the data) or the standard deviation. These measures help us to know how variable the data is. The larger the standard deviation, the more spread out the data is. When outliers are present, it is helpful to only look at the IQR which measures the spread of the middle 50% of the data. This way the outliers don’t exaggerate the variability of the data. I felt that the study we did about the mean absolute deviation (MAD) really helped to show the meaning of a deviation and to introduce the standard deviation.

In week five, we took on the Normal distribution with the Empirical rule and calculating areas under the curve. This is one of my favorite parts of statistics because it allows us to find useful percentages as long as the data is symmetrically distributed. Studying the Empirical Rule (68-95-99.7) helped introduce the idea of percentages of data distributed symmetrically under the normal distribution curve. Since this was only able to give us percentages for the areas between or above/below whole numbers of standard devations from the mean, we needed to also study z-scores and finding areas using these. A z-score tells us how many standard deviations a value is above (positive) or below (negative) the mean. Using a table or technology, we can find the percent of data values that falls above, below, or between z-scores. This can help us determine when a value may be an outlier (outside the 3 standard deviation boundaries or outside the 99.7%). We were also introduced to hypothesis testing in week five. This allowed us to look at a sampling distribution and determine whether or not a sample mean we were looking at was significantly different from the distribution or not in order to reject or fail to reject a hypothesis. When the sample mean had a small enough p-value (or fell in a small enough area on the distribution), it was evident that the hypothesis could be rejected.

In week six, we looked regression and whether or not two quantitative variables were statistically related. We plotted the data on a scatterplot to see if there seemed to be a linear relationship, either positive or negative. We also calculated the correlation coefficient in order to determine how strong the relationship was. We discussed that, even if there was a strong relationship between two variables, this did not necessarily mean that one variable CAUSED the other to respond a certain way. There could be lurking variables that one would need to examine to determine if some other factors are involved. However, the regression analysis did give us a chance to determine if there was a relationship that needed to be examined further. We also plotted the line of least squares in order to predict values of the y variable from a given x variable. All of this can be extremely helpful when analyzing two quantitative variables.

All in all, I feel that I have a great deal of activities that I can pull from with my future statistics students. It is helping me to be the student again and to delve into these concepts from different perspectives.

PART 2—WEEKLY REFLECTION

· Focus on what you learned that made an impression, what may have surprised you, and what you found particularly beneficial and why. Specifically:

· What did you find that was really useful, or that challenged your thinking?

· What are you still mulling over?

· Was there anything that you may take back to your classroom?

· Is there anything you would like to have clarified?

Your Weekly Reflection will be graded on the following criteria for a total of 5 points:

· Reflection is written in a clear and concise manner, making meaningful connections to the investigations & objectives of the week.

· Reflection demonstrates the ability to push beyond the scope of the course, connecting to prior learning or experiences, questioning personal preconceptions or assumptions, and/or defining new modes of thinking.

Your reflection should be about 250 words and no more than 300 words.

INFORMATIONS TO USE TO DO THE ASSIGNMENT IS BELOW

Introduction & Goals

This week’s investigations continue to study the concept of association by looking at bivariate categorical data. Categorical association is less easy to define, particularly when the categorical variables take on multiple values. Investigation 1 develops an intuitive sense of association in a two-way table and looks at ways to represent the association graphically through conditional distributions. Investigation 2 introduces the Chi Square Test for Independence, a formal hypothesis test that produces a likelihood that the variables are independent (not associated).

Goals

Create and interpret a two way table, marginal distribution, and conditional distribution for two categorical variables

Describe the correlation of two categorical variables using marginal distributions

Determine the Chi Square Statistic for two categorical variables

Perform a Test for Independence for two categorical variables and interpret its meaning

Analyze misleading representation of data

Finalize work on Comparative Study Final Project

DoW #7: Depression & Cancer

In week 1, you watched a YouTube video, The Lurking Variable. This video looked at a possible lurking variable in a hypothetical study on Depression and Cancer. The hypothetical data from this video is this week’s DoW. A hypothetical health insurance organization has anecdotal evidence that suggests Depression could be a cause of Cancer. To investigate this claim, they randomly selected 2070 members of their organization. They studied them across their lifespans, recording whether or not the individuals were ever diagnosed with depression or cancer. The results of this hypothetical study are shown in the two-way table below. The health insurance organization would like to know,

“If you experience depression, are you more likely to be diagnosed with cancer?”

The Excel File, Depression Cancer, contains the data for DoW #7.

	Had Cancer?
	Yes	No
Had Depression?	Yes	170	100
	No	50	850

Investigation 1: Categorical Association

This investigation's activities looks at ways to represent and identify association among two categorical variables.

Inv 1, Activity A: Two-Way Tables & Marginal Distributions

The first step we take in analyzing bivariate categorical data is to organize the data in a two-way table. A two-way table shows breaks down the distribution for each category of each variable.

Exercise A1: Complete the Two-Way for DoW #7 by filling in the empty cells for the totals for each variable. (You can do this by hand or using the formulas in the spreadsheet.)

	Had Cancer?	TOTAL
	Yes	No
Had Depression?	Yes	170	1000
	No	50	850
TOTAL

Exercise A2: Reading the Two-Way Table for DoW #7

What are the variables?

What proportion of the participants experienced Depression? What proportion did not?

What proportion of the participants experienced Cancer? What proportion did not?

Do you see a relationship in this two-way table?

Your answers to Exercise A2 allow us to make a Marginal Distribution Table and Graph for each variable.

The Marginal Distribution shows the proportions for each outcome of each variable.

Example: The Table and Bar Graph below show the marginal distribution for the variable “Had Depression” in DoW #7:

Marginal Distribution for “Had Depression”
YES	1170/2070 .565 56.5%
NO	900/2070 .435 43.5%

Exercise A3: Complete the Marginal Distribution Table and Graph for the variable “Had Cancer” in DoW #7 by hand in your journal.

Marginal Distribution for “Had Cancer”
YES
NO

Exercise A4: Interpret the Marginal Distributions. Make two summary statements interpreting the Marginal Distributions for each variable.

Inv 1, Activity B: Conditional Distributions for Each Variable

One way of seeing possible relationships between two variables is to look at the Conditional Distribution for each variable. A conditional distribution, looks at the distribution of one categorical variable within each of the categories of the other variable.

Example: Create a conditional distribution for the variable Had Depression.

Step 1: First, look at the column, Had Cancer? Yes. Calculate the proportion of those with cancer who also had depression. Then calculate the proportion of those with cancer who did not have depression. Notice that these proportions add up to 1 because they make up all of the people who had cancer.

Conditional Distribution for “Had Depression”

	Had Cancer?
	Yes	No
Had Depression?	Yes	170/220 0.773
	No	50/220 0.227
TOTAL	1

From this, we see that if a patient had cancer, he/she had a 77% probability of also having depression.

Step 2: Look at the column, Had Cancer?No. Calculate the proportion of those who did not have cancer, that had depression. Then, calculate the proportion of those who did not have cancer that did not have depression.

Conditional Distribution for “Had Depression”

	Had Cancer?
	Yes	No
Had Depression?	Yes	170/220 0.773	1000/1850 0.541
	No	50/220 0.227	850/1850 0.459
TOTAL	1	1

From this, we see that if a patient did not have cancer, he/she had a 54% probability of having depression. Depression is more evenly distributed for patients without cancer than for patients with cancer.

Step 3: Make a segmented bar graph, using the proportions in the Conditional Distribution Table.

A video clip for this can be found here.

A tutorial can be found here.

Exercise B1: Complete the Conditional Distribution Table for the variable Had Cancer. The first entry has been done for you.

	Had Cancer?
	Yes	No	TOTAL
Had Depression?	Yes	170/1170 0.145		1
	No			1

Exercise B2: Create the graph for the Conditional Distribution for Had Cancer?

Exercise B3: Interpret this conditional distributions for Had Depression? and Had Cancer?. Make at least two summary statements based on each distribution.

Post your response to Exercise B3 to your group’s discussion board for DoW #7 by Wednesday, 10 PM EST.

Review the posts of your group. Make at least two meaningful responses by Friday, 10 PM EST. Pay particular attention to how the interpretations for the two conditional distributions compare to each other. Do you see the same relationship in each graph? Why or why not?

What MIGHT a strong association look like in a two-way table?

	Had Cancer?
	Yes	No
Had Depression?	Yes	a	b
	No	c	d

If there were an association in these categorical variables, how would a, b, c, and d be related? Here are two possible STRONG associations:

“Depression-Yes” to be associated with “Cancer-Yes” and “ Depression-No” to be associated with “Cancer-No”. Thus we would expect the values of a and d to be high and the values of b and c to be low

“Depression-Yes” to be associated with “Cancer-No” and “Depression-No” to be associated with “Cancer-Yes”. Thus we would expect the values of b and c to be high and the values of a and d to be low

We can also think of several situations that would support no association between the variables. For example, a and b could be large with c and d small, or these values could be switched, or all cell counts could be approximately equal1.

1Adapted from http://courses.ncssm.edu/math/Stat_Inst /PDFS/Categorical%20Data%20Analysis.pdf

Exercise B4: The following tables show hypothetical conditional distributions for the variable “Had Depression?”.

Describe the relationship in each distribution, if any exists at all.

Also decide if it is strong or weak.

(a)

	Had Cancer?
	Yes	No
Had Depression?	Yes	100%	0
	No	0	100%

(b)

	Had Cancer?
	Yes	No
Had Depression?	Yes	50%	50%
	No	50%	50%

(c)

	Had Cancer?
	Yes	No
Had Depression?	Yes	20%	75%
	No	80%	25%

(d)

	Had Cancer?
	Yes	No
Had Depression?	Yes	20%	20%
	No	80%	80%

Investigation 2: The Chi Square Test for Independence

In Activity B, we started to think about how the values in the two-way table can be used to identify the relationship among the variables. In this investigation, we will explore the concepts behind a hypothesis test which tests for an association in a two-way table, called the Chi Square Test for Independence.

Inv 2, Activity C: What If There Was No Association?

Recall from the marginal distribution for Had Depression that 57% of the sample of 2070 people had depression and 43% did not have depression.

Marginal Distribution for “Had Depression”
YES	57%
NO	43%

Let’s suppose for a moment that there was no association at all between Cancer and Depression. In that case, it would make sense to think that of those with cancer, 57% would have depression and 43% would not have depression. Likewise, of those who do not have cancer, 57% would have depression and 43% would not have depression.

Exercise C1: Assume there is no association at all between cancer and depression, so that 57% of all people have depression while 43% do not (regardless of their cancer status). Complete the table below to calculate the expected values for each combination.

Expected Values if NO ASSOCIATION between cancer and depression
	Had Cancer?
	Yes (220)	No (1850)
Had Depression?	Yes (57%)	0.57(220) 125.4
	No (43%)

Exercise C2: Compare the expected values you found above with the actual values in the two-way table for DoW #7. Are there significant differences between the actual data, and what we might expect if there was truly no association? In other words, does the 57:43 ratio for depression hold true, no matter whether a patient has cancer or not?

	Had Cancer?
	Yes	No
Had Depression?	Yes	170	1000
	No	50	850

Inv 2, Activity D: The Chi-Square Test for Independence

The Chi-Square Test for Independence is a hypothesis test that makes the same comparison you made in Activity D. It compares what we might expect the data to look like if there was no association to what the actual data is. We then determine the likelihood of this difference, and use that to gauge whether or not there is an association and how likely that association is.

It is called a test for independence, because independence means “no association”. The null hypothesis for this test is that the variables are independent – they have no association:

H0: The Two Variables are Independent

The alternative hypothesis for this test is that the variables are not independent – there is an association:

HA: The Two Variables are not Independent

Recall from Week 5 that in a hypothesis test, we calculate the probability of our observed results under the null hypothesis. If this probability is too small (below the alpha level, which we set at 0.5%), we reject the null hypothesis in favor of the alternative hypothesis. This is the same process for the Chi Square Test for Independence.

The statistic that we use for this test is called the Chi Square Statistic. It is found by comparing the actual observed data to the values we would expect if the variables were independent. For each cell in the two-way table, we calculate the value:

Then, we add them all up to make the Chi Square Statistic.

For DoW #7, you calculated the observed values in Exercise C1:

Observed

	Had Cancer?
	Yes	No
Had Depression?	Yes (57%)	170	1000
	No (43%)	50	850

Expected

	Had Cancer?
	Yes	No
Had Depression?	Yes (57%)	125.4	1054.5
	No (43%)	94.6	795.5

Exercise D1: Calculating the Chi Square Statistic for DoW #7:

(a) Calculate for each cell and confirm the values shown in the table below.

	Had Cancer?
	Yes	No
Had Depression?	Yes (57%)	15.8625199	2.81673779
	No (43%)	21.0270613	3.73381521

(b) Add up the four cells and confirm that the Chi-Square Statistic for DoW #7 is 43.13.

The next step in the Chi Square Test is for us to answer the question: what is the probability of getting a chi square statistic as high as 43.13, if the variables are truly unrelated (independent)?

Exercise D2: Looking at our work and considering previous hypothesis tests, how unlikely is it that we would observe a chi square of 43.13 for DoW #7, if there truly was no association between depression and cancer?

Let’s review what we have done so far with the Chi Square Hypothesis Test for Independence for DoW #7: Null Hypothesis:

H0 = Having Depression and Having Cancer are independent

Alternative Hypothesis:

HA = Having Depression and Having Cancer are NOT independent.

Test Statistic:

Chi Square = sum of =43.13

Level of Significance: alpha = 0.5%

p-value:???

In Exercise D2, we see that it is highly unlikely to obtain a chi square of 43.13 in a situation like DoW #7. We can use the graphing calculator to determine the probability of such a chi square occuring: [2nd][VARS] > c2cdf c2cdf (Test Statistic, 10000, degrees of freedom) Where Degrees of Freedom (d.f.) = (# of rows -1)(# of columns – 1).

For DoW #7, this would be c2cdf (43.13, 10000, 1) = 5.12x10-11. This is the p-value, the probability of obtaining a chi square of 43.13 in a situation like our (with 1 degree of freedom) when the variables are independent. This p-value is much less than our level of significance, 0.5%. As a result, we will reject the null hypothesis, in favor of the alternative hypothesis, that the depression and cancer are related, at the 0.5% significance level.

This may help: https://www.youtube.com/watch?v=j4_SuB4qzWE

OR: http://www.excelfunctions.net/Excel-Chitest-Function.html

Exercise D3: Test for Independence for DoW #7:

The p-value (the probability of getting a chi square as large as 43.13 with independent variables) is less than 0.0001. This is well below the level of significance we set (0.5%). Do we reject the null hypothesis in favor of the alternative hypothesis, that is, is there a relationship between Depression and Cancer at a statistically significant level of 0.5%?

Exercise D4: The Chi Square Test allows you to determine the there is an association between the variables, but does not specify what that relationship is.

What do you think is the association that the chi square test supports in DoW #7?

Explain why this association does NOT support the claim the Depression Causes Cancer, even though we have strong statistical evidence supporting the association. (You may want to revisit the YouTube video, The Lurking Variable from Week 1.) Post your response to Exercise D4 to your group’s DB for DoW #7 by Friday, 10 PM EST. Review the responses of your group. Make at least two follow up posts by Sunday, 10 PM EST.

Comparative Study Draft & Peer Review

Post the draft of your final project to the Comparative Study Peer Review Forum by Sunday, 10 PM EST. Your draft should be in powerpoint format, with your name and the title of your project in the subject line. Your draft will be peer-reviewed by two of your colleagues (NOT in your usual work group). You may want to include a note if there are particular questions you’d like the reviewers to address or areas you would like the reviewers to focus on. This draft will not be graded. You will receive credit for submitting the draft but not for the content. NOTE: THIS IS NOT posted to your comparative study group – it is posted to the whole class. This will give everyone a chance to see each other’s projects, and you will receive peer feedback from two people. Complete Peer Reviews for Two Projects by Week 8, Wednesday, 10 PM EST. The peer review is an opportunity to review the work of your peers, provide feedback to them, and gain insights for your own project. It also is an opportunity to apply your statistical critique skills on a topic other than your own. You will be assigned two projects to review by your instructor. Review each study and provide meaningful feedback to the researcher using the peer review form.

Post your reviews to the Discussion Board by replying to the thread for that project. You will be graded on the content and quality of your feedback in the peer reviews, according to the Comparative Study Peer Review Rubric. It is important that your feedback be timely, in order for others to incorporate the feedback into their work by the project due date.