Psy 202 Week 4 Notes
Psy 202 Week 4 Notes Psy 202
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This 8 page Class Notes was uploaded by Anna Ballard on Friday September 23, 2016. The Class Notes belongs to Psy 202 at University of Mississippi taught by Mervin R Matthew in Fall 2016. Since its upload, it has received 3 views. For similar materials see Elementary Statistics in Psychology at University of Mississippi.
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Date Created: 09/23/16
Lecture 9 Test 2 Lecture Notes 9/20/16 Ch. 6 –> Describing the Relationship between 2 Quantitative Variables: Correlation INTRO • When dealing with multiple values ex: internet explorer vs. murder rate –> correlation ≠ causation - Spurious relationship – 2 variables looking like there is a relationship but the relationship doesn’t actually exist • Looking at 1 variable might not be enough… - With single variable, only info you have is central tendency - With multiple variables… more info which means you can compare variables to each other and see a relationship that will make better predictions (bivariate analysis) - Bivariate analysis can make better predictions than univariate analysis because they help determine next variable results when looking at one is not enough. The Correlational Model (Ch. 6) • # of 80 year olds passing away vs. little girls eating ice-cream - Correlation of 0.00 –> no relation o As X changes, Y stays the same • # of 80 year olds passing away increases and little girls eating ice-cream decreases - Correlation of -1.00 –> perfect negative correlation • # of 80 year olds passing away decreases and little girls eating ice-cream increases - Correlation of +1.00 –> perfect positive correlation • if X increases as Y increases sometimes, there is a correlation of below +1.00 - Strong positive correlation ** Relationship between little girls eating ice-cream and old people dying: - Not direct causal - Another variable (Z) affects X & Y… Y being possibly heat - Correlation ≠ causation o Only say that there is not relationship The Regression Model (Ch. 7) • Quasi-experimental designs • Sleep v. exam grades - Decreased sleep, decreased grades - Increased stress and decreased grades with possibly decreased sleep Correlation Model (continued) • Quantitative Data –> use interval or ratio scores • Moving from univariate v. bivariate dsitrbutions - Bivariate: 2 sets of scores Univariate X F(X ) 6 0 5 0 4 0 3 2 2 2 1 2 Bivariate X Y F(X, Looking at X vs. Y pairs Y) - care because you get frequency of each pair 2 3 1 2 2 1 1 1 3 1 2 1 1 0 Scatter Plot • Assessing the strength of linear association - interval/ratio scores –> must be able to imagine a line o regression line – imaginary line through scatter plot the closer the points, the stronger the relationship - Below: positive correlation o If imagined line is flipped in other direction – negative correlation o If no imagined line – no correlation Early Work • Moving from variance to Covariance to Correlation - Covariance by itself is not too helpful o Range of X and range of Y Standardize covariance Issues in Interpretation: Magnitude • increased magnitude of the correlation coefficient means there is a more consistent relationship - Does not say how or why changes but tells us if it does often Issues in Interpretation: Linearity • Drinks some Red Bull, increased concentration • Drinks rest of red bull, decreased concentration • Relationship nonlinear –> X increases and Y increases but after so long X decreases - If plot is non linear, the correlation coefficient underestimates the relationship between the variables Lecture 10 Test 2 Notes 9/21/16 Ch. 6 –> Describing Relationship Between 2 Quantitative Variables: Correlation • If you don’t have linearity, the correlation coefficient doesn’t work • Besides linearity, take into account: - Restrictions in Range – Restricted on either X or Y values o Makes correlation look like there’s no relationship when there is - Presence of Outliers – could make the relationship look weaker or stronger than it actually is o Correlation coefficient tells us typically how far points are from the regression line and outliers skew that regression line Flatter the regression line, weaker the relationship - Presence of Subgroups – strengthen correlation coefficient o Look at groups separately – no correlation, but groups together – correlation Ex: gender gaps - Causality – how relevant the 2 variables are to each other o Ex: president vs. gas prices… George Bush v. Obama, gas prices went down when Obama came into office but he is not strictly the reason those prices went down o Know how to find bias when causal inferences aren’t true • How R helps us with this… - Boxplot: little circles: outliers (throw off correlation coefficient) - Scatterplot: tells us if correlation is linear or not - Sunflower plot: red lines means points are repeating - Correlation coefficient: how strong (closer to |1|) the correlation is Other Product-Moment Correlation Coefficients • Continuous and linear data is necessary (interval and ratio scales) • Spearman Rank Correlation – > 2 sets of ordinal scores (rank info, not distance info) r ranks= Difference in 2 Rank G∑D i n(n –2 • Point-Biserial Correlatioa Difference between sample r b Y – 1 means √ (n -n )÷ (n +n )( n Y2 1 2 1 2 1 S y Estimated Standard Deviation of Parent Population Yes No Male 7 14 Femal 21 9 e • The Phi Coefficient - 2 sets of categorical scores - if more than 2 possible outcomes, this will not work **Dichotomous Scores** r = bc – ad ø √(a+c)(b+d)(a+b) Non-Product-Moment Correlations (c+d) non-product-moment correlation with quantitative and dichotomous variables o continuous scores treated as categorical Bi-serial Correlation Coefficient • Treat one set as dichotomous - both continuous but one treated as dichotomous Proportion in Group 1 rbis= Y –1 Height of ( Pq ÷ y ) Distribution 2 S y >10 ≤ 10 Group 0ion in Bi-serial Coefficient 20 7 14 Tetra-choric Correlation • Both treated as dichotomous • Treat one set as dichotomous ≤ 20 21 9 R tet = cos (180 ÷ (1 + √(bc+ad))) Non-Product-Moment Correlation for Categorical Variables Contingency coefficient - more than 2 categories - harder to interpret because it doesn’t work with correlation coefficient - like a chi-square problem 2 2 C = √(X ÷ (n+ X )) Cramer’s V - bound between -1 and +1 - easier to interpret because correlation coefficient is relatively similar 2 V = √[X ÷ (n(q - 1))] Lecture 11 9/23 Ch. 7 –> Describing the Relationship between Two Quantitative Variables: Regression In Ch. 6… • explaining relationship between 2 different variables - correlation mode - regression model o still looking at relationship between 2 different quantitative variables but also looking at some causal variable amount of sleep, stress, and grades independent variables = predictors dependent variable = criterion Regression Model • Telling “how” in addition to “if” - all in correlation of |1| but slopes of lines are different - as X goes up, how much does Y go up –> slope tells us this REVIEWING THE ALGEBRA OF LINES Y = ƒ(x) = mx + b - slope = m –> how much y changes as x changes - y-intercept = b –> value of y when x=0 Moving from Algebraic Notation to Statistical Notation Y (hat) = ƒ(x) = b + b X 0 1 - y-hat –> predicted value of y - b0–> y-intercept - b1–> slope • If you know information about the predictor, it will give you information about the criterion Simple Approach • Scatter plot - connect dots to find regression line o not a single smooth line, a bunch of lines create jagged line - create 1 line that gets as close to many points as possible o real data is a little more complicated figure out which ones don’t work then ones that do • For most scatter plots, there are infinite number of lines that could summarize it - so how to find which line is the best? This old idea –> squared errors - want squared errors to be small - smallest amount of space between prediction and what actually happens - mean gives the smallest squared errors (a = µ), giving us better predictions Regression Analysis: A Visual, Conceptual Approach 2 SS error = ∑(Y – Y(hit)) i - error of estimation squared o vertical distance between Y and Y hat i i Yihat tells us where Y should be along the regression line • Find the line that minimizes error Y (hat) = ƒ(x) = b + b X 0 1 - Yi= 1 + 2Xi o make number of squared errors as small as possible I X Y Yhat Y-Yhat (Y- Yhat)2 1 1 3 3 0 0 2 1 2 3 -1 1 3 2 4 5 -1 1 4 2 5 5 0 0 5 3 5 7 -2 4 6 3 6 7 -1 1 ∑ 12 25 30 15 7 I X Y X2 Y2 XY 1 1 3 1 9 3 2 1 2 1 4 2 3 2 4 4 16 8 4 2 5 4 25 10 5 3 5 9 25 15 6 3 6 9 36 18 ∑ 12 25 28 15 56 The Mathematical Approach: Finding the Slope B = 1.5 1 B0= Y(bar) – 1 X(bar) - Y (bar) –> sample mean of Y - X (bar) –> sample mean of X Yihat) = 1.167 +1.5Xi(measure of central tendency) - Both numbers minimizes SSerrorr this data set The Mathematical Approach: The Standard Error of the Estimate 2 SE est = √∑[(Y – Y (hat)) ÷ (n-2)] - Lose 2 degrees of freedom for estimates of 2 means - ESest 0.677 o What is this answer telling you? Average distance between predicted and observed scores (measure of variability)
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