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Bivariate Analyses

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Bivariate Procedures I Overview Chi-square test Chi-square test T-test T-test Correlation Correlation

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Chi-Square Test Relationships between nominal variables Relationships between nominal variables Types: Types: 2x2 chi-square 2x2 chi-square Gender by Political Party Gender by Political Party 2x3 chi-square 2x3 chi-square Gender by Dosage (Hi vs. Med. Vs. Low) Gender by Dosage (Hi vs. Med. Vs. Low)

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Starting Point: The Crosstab Table Example: Example: Gender (IV) MalesFemales Democrat120 Party (DV) Republican102 Total1122

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Column Percentages Gender (IV) MalesFemales Democrat9%91% Party (DV) Republican91%9% Total100%100%

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Row Percentages Gender (IV) MalesFemalesTotal Democrat5%95%100% Party (DV) Republican83%17%100%

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Full Crosstab Table MalesFemalesTotal Democrat %95% 9%91%64% Republican %17% 91%9%36% Total %67%100%

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Research Question and Hypothesis Research Question: Research Question: Is gender related to party affiliation? Is gender related to party affiliation? Hypothesis: Hypothesis: Men are more likely than women to be Republicans Men are more likely than women to be Republicans Null hypothesis: Null hypothesis: There is no relation between gender and party There is no relation between gender and party

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Testing the Hypothesis Eyeballing the table: Eyeballing the table: Seems to be a relationship Seems to be a relationship Is it significant? Is it significant? Or, could it be just a chance finding? Or, could it be just a chance finding? Logic: Logic: Is the finding different enough from the null? Is the finding different enough from the null? Chi-square answers this question Chi-square answers this question What factors would it take into account? What factors would it take into account?

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Factors Taken into Consideration Factors: Factors: 1. Magnitude of the difference 1. Magnitude of the difference 2. Sample size 2. Sample size Biased coin example Biased coin example Magnitude of difference: Magnitude of difference: 60% heads vs. 99% heads 60% heads vs. 99% heads Sample size: Sample size: 10 flips vs. 100 flips vs. 1 million flips 10 flips vs. 100 flips vs. 1 million flips

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Chi-square Chi-Square starts with the frequencies: Chi-Square starts with the frequencies: Compare observed frequencies with frequencies we expect under the null hypothesis Compare observed frequencies with frequencies we expect under the null hypothesis

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What would the Frequencies be if there was No Relationship? MalesFemalesTotal Democrat21 Republican12 Total112233

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Expected Frequencies (Null) MalesFemalesTotal Democrat71421 Republican4812 Total112233

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Comparing the Observed and Expected Cell Frequencies Formula: Formula:

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Calculating the Expected Frequency Simple formula for expected cell frequencies Simple formula for expected cell frequencies Row total x column total / Total N Row total x column total / Total N 21 x 11 / 33 = 7 21 x 11 / 33 = 7 21 x 22 / 33 = x 22 / 33 = x 11 / 33 = 4 12 x 11 / 33 = 4 12 x 22 / 33 = 8 12 x 22 / 33 = 8

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Observed and Expected Cell Frequencies MalesFemalesTotal Democrat Republican Total112233

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Plugging into the Formula O - E SquareSquare/E Cell A = 1-7 = /7 = 5.1 Cell B = = 63636/14 = 2.6 Cell C = 10-4 = 63636/4 = 9 Cell D = 2-8 = /8 = 4.5 Sum = 21.2 Chi-square = 21.2

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Is the chi-square significant? Significance of the chi-square: Significance of the chi-square: Great differences between observed and expected lead to bigger chi-square Great differences between observed and expected lead to bigger chi-square How big does it have to be for significance? How big does it have to be for significance? Depends on the “degrees of freedom” Depends on the “degrees of freedom” Formula for degrees of freedom: Formula for degrees of freedom: (Rows – 1) x (Columns – 1)

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Chi-square Degrees of Freedom 2 x 2 chi-square = 1 2 x 2 chi-square = 1 3 x 3 = ? 3 x 3 = ? 4 x 3 = ? 4 x 3 = ?

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Chi-square Critical Values dfP = 0.05P = 0.01P = * If chi-square is > than critical value, relationship is significant

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Chi-Square Computer Printout

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Multiple Chi-square Exact same procedure as 2 variable X 2 Exact same procedure as 2 variable X 2 Used for more than 2 variables Used for more than 2 variables E.g., 2 x 2 x 2 X 2 E.g., 2 x 2 x 2 X 2 Gender x Hair color x eye color Gender x Hair color x eye color

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Multiple chi-square example

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The T-test Groups T-test Groups T-test Comparing the means of two nominal groups Comparing the means of two nominal groups E.g., Gender and IQ E.g., Gender and IQ E.g., Experimental vs. Control group E.g., Experimental vs. Control group Pairs T-test Pairs T-test Comparing the means of two variables Comparing the means of two variables Comparing the mean of a variable at two points in time Comparing the mean of a variable at two points in time

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Logic of the T-test A T-test considers three things: A T-test considers three things: 1. The group means 1. The group means 2. The dispersion of individual scores around the mean for each group (sd) 2. The dispersion of individual scores around the mean for each group (sd) 3. The size of the groups 3. The size of the groups

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Difference in the Means The farther apart the means are: The farther apart the means are: The more confident we are that the two group means are different The more confident we are that the two group means are different Distance between the means goes in the numerator of the t-test formula Distance between the means goes in the numerator of the t-test formula

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Why Dispersion Matters Small variances Large variances

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Size of the Groups Larger groups mean that we are more confident in the group means Larger groups mean that we are more confident in the group means IQ example: IQ example: Women: mean = 103 Women: mean = 103 Men: mean = 97 Men: mean = 97 If our sample was 5 men and 5 women, we are not that confident If our sample was 5 men and 5 women, we are not that confident If our sample was 5 million men and 5 million women, we are much more confident If our sample was 5 million men and 5 million women, we are much more confident

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The four t-test formulae 1. Matched samples with unequal variances 1. Matched samples with unequal variances 2. Matched samples with equal variances 2. Matched samples with equal variances 3. Independent samples with unequal variances 3. Independent samples with unequal variances 4. Independent samples with equal variances 4. Independent samples with equal variances

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All four formulae have the same Numerator Numerator X1 - X2 (group one mean - group two mean) X1 - X2 (group one mean - group two mean) What differentiates the four formulae is their denominator What differentiates the four formulae is their denominator denominator is “standard error of the difference of the means” denominator is “standard error of the difference of the means” each formula has a different standard error each formula has a different standard error

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Independent sample with unequal variances formula Standard error formula (denominator): Standard error formula (denominator):

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T-test Value Look up the T-value in a T-table (use absolute value ) First determine the degrees of freedom ex. df = (N1 - 1) + (N2 - 1) = 70 For 70 df at the.05 level =1.67 ex > 1.67: Reject the null (means are different)

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Groups t-test printout example Groups t-test printout example

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Pairs t-test example

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Pearson Correlation Coefficient (r ) Characteristics of correlational relationships: Characteristics of correlational relationships: 1. Strength 1. Strength 2. Significance 2. Significance 3. Directionality 3. Directionality 4. Curvilinearity 4. Curvilinearity

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Strength of Correlation: Strong, weak and non-relationships Strong, weak and non-relationships Nature of such relations can be observed in scatter diagrams Nature of such relations can be observed in scatter diagrams Scatter diagram Scatter diagram One variable on x axis and the other on the y-axis of a graph One variable on x axis and the other on the y-axis of a graph Plot each case according to its x and y values Plot each case according to its x and y values

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Scatterplot: Strong relationship BOOKREADINGBOOKREADING Years of Education

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Scatterplot: Weak relationship INCOMEINCOME Years of Education

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Scatterplot: No relationship SPORTSINTERESTSPORTSINTEREST Years of Education

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Strength increases… As the points more closely conform to a straight line As the points more closely conform to a straight line Drawing the best fitting line between the points: Drawing the best fitting line between the points: “the regression line” “the regression line” Minimizes the distance of the points from the line: Minimizes the distance of the points from the line: “least squares” “least squares” Minimizing the deviations from the line Minimizing the deviations from the line

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Significance of the relationship Whether we are confident that an observed relationship is “real” or due to chance Whether we are confident that an observed relationship is “real” or due to chance What is the likelihood of getting results like this if the null hypothesis were true? What is the likelihood of getting results like this if the null hypothesis were true? Compare observed results to expected under the null Compare observed results to expected under the null If less than 5% chance, reject the null hypothesis If less than 5% chance, reject the null hypothesis

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Directionality of the relationship Correlational relationship can be positive or negative Correlational relationship can be positive or negative Positive relationship Positive relationship High scores on variable X are associated with high scores on variable Y High scores on variable X are associated with high scores on variable Y Negative relationship Negative relationship High scores on variable X are associated with low scores on variable Y High scores on variable X are associated with low scores on variable Y

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Positive relationship example BOOKREADINGBOOKREADING Years of Education

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Negative relationship example RACIALPREJUDICERACIALPREJUDICE Years of Education

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Curvilinear relationships Positive and negative relationships are “straight-line” or “linear” relationships Positive and negative relationships are “straight-line” or “linear” relationships Relationships can also be strong and curvilinear too Relationships can also be strong and curvilinear too Points conform to a curved line Points conform to a curved line

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Curvilinear relationship example FAMILYSIZEFAMILYSIZE SES

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Curvilinear relationships Linear statistics (e.g. correlation coefficient, regression) can mask a significant curvilinear relationship Linear statistics (e.g. correlation coefficient, regression) can mask a significant curvilinear relationship Correlation coefficient would indicate no relationship Correlation coefficient would indicate no relationship

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Pearson Correlation Coefficient Correlation coefficient Correlation coefficient Numerical expression of: Numerical expression of: Strength and Direction of straight-line relationship Strength and Direction of straight-line relationship Varies between –1 and 1 Varies between –1 and 1

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Correlation coefficient outcomes -1 is a perfect negative relationship -.7 is a strong negative relationship -.4 is a moderate negative relationship -.1 is a weak negative relationship 0 is no relationship.1 is a weak positive relationship.4 is a moderate positive relationship.7 is a strong positive relationship 1 is a perfect positive relationship

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Pearson’s r (correlation coefficient) Used for interval or ratio variables Used for interval or ratio variables Reflects the extent to which cases have similar z-scores on variables X and Y Reflects the extent to which cases have similar z-scores on variables X and Y Positive relationship—z-scores have the same sign Positive relationship—z-scores have the same sign Negative relationship—z-scores have the opposite sign Negative relationship—z-scores have the opposite sign

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Positive relationship z-scores PersonXzYz A B C D E

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Negative relationship z-scores PersonXzYz A B C D E

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Conceptual formula for Pearson’s r Multiply each cases z-score Multiply each cases z-score Sum the products Sum the products Divide by N Divide by N

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Significance of Pearson’s r Pearson’s r tells us the strength and direction Pearson’s r tells us the strength and direction Significance is determined by converting the r to a t ratio and looking it up in a t table Significance is determined by converting the r to a t ratio and looking it up in a t table Null: r =.00 Null: r =.00 How different is what we observe from null? How different is what we observe from null? Less than.05? Less than.05?

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Computer Printout

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