Thinking about correlation

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Thinking about correlation

  1. 1. Thinking about Correlation Chapter 2 – Third Slideshow
  2. 2. Let’s start by revisiting the story of Jen and her son Tyler.
  3. 3. Marie, Jen, and Sara sit and chatat the local park every Saturday. Jen keeps a very close eye on herson, Tyler. She warns him about thegravel under the swings. Then she tellsTyler to be careful at the top of the slide.When all of the kids climb the junglegym, she jumps up and runs over, tellinghim to get off because it is not safe.
  4. 4. Scolded, and feeling sad, Tyler walks over to his mother for a hug. After comforting him, Jen tells him to go play with the children by the swings. Watching this, Sara and Marie exchange adisgusted look. They worry that the way Jen treatsTyler may make him grow up to be gay.
  5. 5. How can we research this question? Can you change a child’s sexuality, or their gender identity, by how you parent them?
  6. 6. Many beliefs include a relationship between two variables.Marie and Sara’s belief is a good example. Their concern that Jen’s parenting may cause Tyler to grow up to be gay involves two variables.
  7. 7. Although Marie and Sara are reacting to the specific things that Jen did with Tyler at the park, their belief is more general. To see the variables, you have to think more abstractly than just this one case.
  8. 8. The two variables are:1. The degree to which a mother is protective of her son.2. The probability of a boy growing up to have a homosexual orientation.
  9. 9. Each variable can vary across a continuum of values.1. The degree to which a mother is protective of her son. Not Protective Very Protective2. The probability of a boy growing up to have a homosexual orientation. Definitely not Homosexual Definitely Homosexual
  10. 10. Marie and Sara are worried that Jen falls high on the first variable, and that Tyler may be high on the second variable because of that.1. The degree to which a mother is protective of her son. Jen Not Protective Very Protective2. The probability of a boy growing up to have a homosexual orientation. Tyler Definitely not Homosexual Definitely Homosexual
  11. 11. Marie and Sara believe that these two variables are related. They believe that the more protective a mother is, the higher the probability that her son will grow up to identify as a gay man.
  12. 12. Even more, they believe that the first variable causes the second variable.They believe that a mother’s protectiveness causes her son to be gay.
  13. 13. As you learned earlier, looking for correlation is the first step in the search for cause and effect relationships. Conditions to establish Cause and Effect. 1. The variables are correlated. 2. The cause comes before the effect. 3. There are no other variables to explain the effect.
  14. 14. Correlations Have Two Ends, or SidesAs you learned earlier, if you think two variables are correlated, be sure to think about both sides of the relationship, the high end and the low end.
  15. 15. Tools to Think About Correlations
  16. 16. ScatterplotsThere are many tools to study correlation, and I encourage you to take a course in statistics to learn more about them. One of these tools is a scatterplot.
  17. 17. ExampleLet’s revisit another example as we introduce this new tool.
  18. 18. Remember, we hypothesized that teacher’s niceness causes his/her students to learn more.Also remember, the first step when looking for a cause is to see if there is a correlation.
  19. 19. For this example, let’s pretend that you formed this belief by watching a particular teacher, Mr. Carter, who was really nice and pleasant. You also noticed how well students did in Mr. Carter’s class. Perhaps it was his niceness that caused this.
  20. 20. Note, this is a belief formed through personal experience (Review: this was one of our ways of knowing!)
  21. 21. Review - Thinking about VariablesThis experience represents a co-occurrence of two things: a nice teacher and a class performing well. Each of those is one possible value on a variable.Seeing a “nice teacher” can be thought of as seeing a person who is high in niceness compared to other teachers (a variable)Seeing a “class that is doing well” can be thought of as a class that is scoring high on a test of learning compared to other classes (a variable).
  22. 22. Interpreting a ScatterplotOn the next slide, each dot represents a particular teacher’s niceness and his/her class’s learning. This graph shows 18 teacher’s and classes’ scores.
  23. 23. Scatterplot Example Mr. Carter’s Class Ms. Stark’s ClassNot at all Nice Very Nice Teacher Niceness
  24. 24. Interpreting a ScatterplotOne variable is represented as horizontal distances. Dots to the right represent nicer teachers; dots to the left represent less nice teachers.
  25. 25. Scatterplot Example Mr. Carter’s Class Ms. Stark’s ClassNot at all Nice Very Nice Teacher Niceness
  26. 26. Interpreting a ScatterplotThe other variable is represented as vertical distances. Dots that are higher represent classes that are performing well; lower dots represent classes that are doing poorly.
  27. 27. Scatterplot Example Mr. Carter’s Class Ms. Stark’s ClassNot at all Nice Very Nice Teacher Niceness
  28. 28. Tools to See Correlations - ScatterplotsYou can see correlations by viewing scatterplots. If the two variables are positively related, you see an oval- shaped cluster of dots that slopes upward, starting low on the left and getting higher to the right, which is what you see below.
  29. 29. Tools to See Correlations - ScatterplotsVariables can also be negatively related. If the two variables are negatively related, you see an oval- shaped cluster of dots that slopes downward, starting high on the left and getting lower to the right, which is what you see below.
  30. 30. Tools to See CorrelationScatterplots are extremely useful tools. However, not everyone finds graphs to be useful. Here is another way to try to think about correlations.
  31. 31. Tools to See Correlation – Two-by-Two TablesWe can use the same observations to categorize Mr. Carter and his excellent class.This is our second tool for thinking about correlations: a two-by-two table.
  32. 32. Two-by-Two Table ExampleBelow, you see a two-by-two table, which has two columns,and two rows (hence the name).
  33. 33. Two-by-Two Table ExampleWe build the table by making columns represent low andhigh levels of one variable – in this case, teacher niceness. Teachers Who are Not Teachers Who are Nice Nice
  34. 34. Two-by-Two Table ExampleWe make the rows represent low and high levels of theother variable – in this case, the degree of StudentLearning in the teacher’s class. Teachers Who are Not Teachers Who are Nice Nice Class Mastering Content Very Little Learning
  35. 35. Two-by-Two Table ExampleOur belief was based on a personal experience with Mr.Carter. That represents one case. His class wouldcontribute one count to the cell shown below (highlightedyellow). Teachers Who are Not Teachers Who are Nice Nice Class Mastering Content 1 Very Little Learning
  36. 36. Two-by-Two Table ExampleThis does not show a correlation, though. This just showsco-occurrence. We have one example of a nice teacherwith a class of students who master the content heteaches. Teachers Who are Not Teachers Who are Nice Nice Class Mastering Content 1 Very Little Learning
  37. 37. Two-by-Two Table ExampleLet’s add Ms. Stark’s class. She was not nice, and herstudents did not learn very much. Her class contributesone count to the cell shown below (highlighted in yellow). Teachers Who are Not Teachers Who are Nice Nice Class Mastering Content 1 Very Little Learning 1
  38. 38. Two-by-Two Table ExampleNext, we go and collect data from an additional 16 classes,and add them to the counts in our two-by-two table. Below,you see an example of what this might look like (these arenot real data!). Teachers Who are Not Teachers Who are Nice Nice Class Mastering Content 2 8 Very Little Learning 7 1
  39. 39. For there to be a correlation, you need to see most cases showing up in two cells diagonal to each other, and very few counts in the other two cells.
  40. 40. Two-by-Two Table ExampleNotice that in this example, most of the cases are in theupper right and lower left cells (yellow). Few cases arecounted in the other two cells. This pattern indicates apositive correlation between teacher niceness and studentlearning. Teachers Who are Not Teachers Who are Nice Nice Class Mastering Content 2 8 Very Little Learning 7 1
  41. 41. Two-by-Two Table ExampleNotice this is the same pattern that we had with thescatterplot for a positive correlation. This isn’t acoincidence. Math is awesome! Low on A High on A High on B Few Lots Low on B Lots Few
  42. 42. Two-by-Two Table, Negative CorrelationYou see a correlation by two diagonal cells having largecounts, and the other two having few cases in them. It canalso happen in the pattern below. This is a negativecorrelation. Low on A High on A High on B Lots Few Low on B Few Lots
  43. 43. Two-by-Two Table, Negative CorrelationAgain, notice the downward slope in the scatterplot is thesame pattern of cells in the two-by-two table. High on Low on A AHigh on B Lots Few Low on B Few Lots
  44. 44. Two-by-Two Table, Negative CorrelationHere is an example of a negative correlation. These arefictional counts based on 100 students. Real data wouldnot be this dramatic. Students Earning Students Earning Poor Grades in Good Grades in Classes Classes Students who Work Full-time Outside of 19 15 Classes Students who Work less than Full-time or 17 49 Do Not Work
  45. 45. Two-by-Two Table, Negative CorrelationNote that the biggest counts are on a diagonal, highlightedin yellow. Also note that the other two cells are lower. Students Earning Students Earning Poor Grades in Good Grades in Classes Classes Students who Work Full-time Outside of 19 15 Classes Students who Work less than Full-time or 17 49 Do Not Work
  46. 46. Two-by-Two Table, Negative CorrelationThis negative correlation means that students who work fulltime tend to do more poorly in their classes. Students Earning Students Earning Poor Grades in Good Grades in Classes Classes Students who Work Full-time Outside of 19 15 Classes Students who Work less than Full-time or 17 49 Do Not Work
  47. 47. Two-by-Two Table, BenefitsTwo-by-two tables are an incredibly useful tool for thinkingabout relationships between variables.Everyday experiences rarely help us observe cases that fitall four cells. This is one of the first benefits ofsystematically collecting evidence with the scientificmethod – we can investigate all four cells to find evidenceof a correlation.
  48. 48. Two-by-Two Table, CategoriesLet’s expand our tool.Another type of variable is one where something is eitherpresent or absent, or a member of a category.
  49. 49. Two-by-Two Table, CategoriesFor example, you can either wear glasses, or not wearglasses. This varies across people.For an example of categories, you could either be left-handed or right-handed. This is also a variable.
  50. 50. Two-by-Two Table, CategoriesPerhaps you hypothesize that right-handed people aremore likely to have glasses.Let’s pretend we purposefully find 100 left-handed and 100right-handed people and count how many have glasses. Left-Handed Right-Handed Has Glasses ? ?Does Not Have Glasses ? ?
  51. 51. Two-by-Two Table, CategoriesPretend you make careful records and get the countsshown below. These data indicate no relationship. Thesetwo variables are uncorrelated. You can see this becauseall of the cells have about the same number of people. Left-Handed Right-Handed Has Glasses 50 49Does Not Have Glasses 50 51
  52. 52. Although I’m suggesting that you think about relationships by using tools such as the scatterplot and two-by-two tables, psychologists use more tools than these.
  53. 53. Specifically, we need a way of determining when to conclude that the variables are “correlated” and when to conclude that there is no relationship. We accomplish this by using statistics.
  54. 54. Without statistics, we can not fully use these tools to help us make decisions about correlations. However, we can make a big step toward thinking more like a psychologist about variables and their relationships.
  55. 55. Let’s look at the example that we started with at the beginning of the slideshow. Marie and Sara believe that a mother’s protectiveness causes her son to be gay.
  56. 56. The first variable is a mother’s protectiveness. Let’s think about this variable for a two-by-two table. Some mothers are extremely protective, and others are normal (meaning they have a typical, or average, level of protectiveness).(Keep in mind that we are using the term “normal” to mean “typical.” Psychologists use this word differently. We don’t mean any judgment of goodness or badness.)
  57. 57. The second variable is the son’s adult sexual orientation. We can think about this as a category, either Homosexual or Heterosexual.
  58. 58. Two-by-Two Table for Sexuality ExampleThe data below are for 1,000 imaginary men. These arefictitious data, but they reflect what real studies have found. Heterosexual Son Homosexual Son Extremely Protective Mother 145 6 Normal Mother 825 24
  59. 59. Two-by-Two Table for Sexuality ExampleUnfortunately, the picture is not immediately clear. Let’slook at what we can conclude. Heterosexual Son Homosexual Son Extremely Protective Mother 145 6 Normal Mother 825 24
  60. 60. Two-by-Two Table for Sexuality Example1. Most men are heterosexual. Notice that there are far more men in that column (highlighted yellow), regardless of row. Heterosexual Son Homosexual Son Extremely Protective Mother 145 6 Normal Mother 825 24
  61. 61. Two-by-Two Table for Sexuality Example2. Most mothers are Normal. Most cases are in the bottom row, reflecting that “normal mothers” are common.So far, these do not tell us anything about a correlation. Heterosexual Son Homosexual Son Extremely Protective Mother 145 6 Normal Mother 825 24
  62. 62. Two-by-Two Table for Sexuality Example3. Most homosexual men did not have overprotective mothers. You can see this in the highlighted column. Heterosexual Son Homosexual Son Extremely Protective Mother 145 6 Normal Mother 825 24
  63. 63. Two-by-Two Table for Sexuality Example4. Most men with overprotective mothers are heterosexual. You can see this in the top row below.This means that knowing Jen is overprotective, we would still predict that Tyler will be heterosexual, because most men with overprotective mothers are heterosexual. Heterosexual Son Homosexual Son Extremely Protective Mother 145 6 Normal Mother 825 24
  64. 64. Conclusion for Marie and SaraData such as these are not consistent with Marie and Sara’s belief. If a mother’s protectiveness causes her son’s homosexuality, we would see the pattern for a correlation.
  65. 65. Conclusion for Marie and SaraFurthermore, if a mother’s protectiveness was the cause of homosexuality in men, as some people believe, then we should see a perfect correlation.The next slide shows what a perfect correlation would look like for this example.
  66. 66. Example of Perfect CorrelationAs before, there are far more heterosexual men thanhomosexual men. If there is a perfect correlation, allheterosexual men would have normal mothers and allhomosexual men would have overprotective mothers. Heterosexual Son Homosexual Son Extremely Protective Mother 0 30 Normal Mother 970 0
  67. 67. Example of Perfect CorrelationNotice the pattern for the strong correlation:The diagonal has all cases (or most), and the other two cells have none (or few). Heterosexual Son Homosexual Son Extremely Protective Mother 0 30 Normal Mother 970 0
  68. 68. Summary – Seeing Correlations To show a correlation, psychologists use statistics. You can use a scatterplot to see a correlation.
  69. 69. Summary – Seeing Correlations with Two-by-Two Tables You can also use a two-by-two table to think about correlations.  Personal experiences typically only offer us one cell of the two-by-two table (co-occurrence).  For a strong correlation to exist, two diagonal cells have to have most cases, and the other two cells need to have few cases.
  70. 70. What does this mean for Marie and Sara’s belief?
  71. 71. Across decades of research, psychologists have been unable to find any one type of parenting or any activity that seems to cause homosexuality.
  72. 72. Although the belief that parents affect their children’s sexuality has tenacity(many people continue to believe it), it is not supported by evidence.
  73. 73. Some religious authorities believe that parents have a moral obligation to behave in certain ways. Scientific evidence has nothing to say about this. We can not study ultimate concerns such as one’s salvation or moral standing with a deity. These are supernatural questions, outside of what science can study.
  74. 74. However, if an authority suggests that a type of parenting will lead to a child becoming homosexual (a claim that we can study with the scientific method), then that claim is inconsistent with empirical evidence.
  75. 75. Practice identifying variables.Practice trying to identify both sides of a correlation (e.g., high side: nice teachers have high performing classes, and low side: mean teachers have low performing classes).Practice trying to think about two variables in a scatterplot, or all four cells in a two-by-two table for a correlation.

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