1. Advanced Statistical Methods: Continuous Variables http://statisticalmethods.wordpress.com http://statisticalmethods.wordpress.com REVIEW Dr. Irina Tomescu-Dubrow tomescu.1@sociology.osu.edu

2. Descriptive & Inferential Statistics -Descriptive statistics describe samples of subjects in terms of variables or combination of variables; -Inferential statistics test hypotheses (about differences in population(s)/ relationships) on the basis of measurements made on samples of subjects

3. Basic Characteristics of A Distribution (continuous variable) Mean = (Σ X i ) / N Standard deviation & Variance ____ ____ S = √ Var = √ s 2 VAR = s 2 = ∑ (X i – Mean of X) 2 / N

4. Shape Three properties of the distribution’s shape (metric variables): MODALITY KURTOSIS SKEWNESS

5. Skewness -Mean > Median > Mode  positively skewed distribution. -Mean < Median < Mode  negatively skewed distribution. -Mean = Median = Mode  normal distribution.

6. Normal Distribution –Unimodal –Mesokurtic (a moderate peakedness) –Symetric (zero skewness) Carl Friedrich Gauss, 1794

7. Normal distribution Every normal curve (regardless of its mean or standard deviation) conforms to the following rule: ~ 68% of the area under the curve falls within 1 standard deviation of the mean. ~ 95% of the area under the curve falls within 2 standard deviations of the mean. ~ 99.7% of the area under the curve falls within 3 standard deviations of the mean.

8. Normal (probability) distribution

9. Z-scores For finding points in the normal distribution the mean value and the standard deviation are crucial. Z score = (Value – Mean) / Standard deviation Definition: a z-score measures the difference between a raw value (a variable value X i ) and the mean, using the standard deviation of the distribution as the unit of measure. ___ Z score (X i ) = X i – Mean for X / √ s 2 The numerical value of the z-score specifies the distance from the mean expressed in terms of the proportion of the standard deviation.

10. Raw values & Z-scores 1. Shape: the shape of the z-score distribution (i.e. standard distribution) is the same as the shape of the raw-score distribution. 2. The mean: when normally distributed raw scores are transformed into z-scores, the resulting z-score distribution will always have a mean = zero. This fact makes the mean a convenient reference point. 3. The standard deviation: when normally distributed raw scores are transformed into z-scores, the resulting z-score distribution will always have a standard deviation = one.

11. Raw scores for education in 2003, POLPAN data Z-scores for education (i.e. standradized variable), POLPAN data

12. Bivariate Correlation and Regression Assumptions: -Random sampling -Both variables = continuous (or can be treated as such) -Linear relationship between the 2 variables; -Normally distributed characteristics of X and Y in the population

13. Linear Relationship The line = a mathematical function that can be expressed through the formula Y = a + b*X, where Y & X are our variables. Y, the dependent variable, is expressed as a linear function of the independent (explanatory) variable X.

14. In reality data are scattered

15. For z scores

16. Correlation - measures the size & the direction of the linear relation btw. 2 variables (i.e. measure of association) - unitless statistic (it is standardized); we can directly compare the strength of correlations for various pairs of variables The stronger the relationship between X & Y, the closer the data points will be to the line; the weaker the relationship, the farther the data points will drift away from the line. Pearson’s r = the sum of the products of the deviations from each mean, divided by the square root of the product of the sum of squares for each variable. If X and Y are expressed in standrad scores (i.e. z-scores), we have Z(y) = β*Z(x) and r = Σ(Zy*Zx)/N = beta

17. Interpretation of Pearson’s r: 1. Strength of relationship: correlations are on a continuum; any value of r btw. –1 & 1 we need to describe: weak  |0.10| moderate  |0.30| strong  |0.60| 0 = no relationship 2. Direction of relationship: a) Positive correlation: whenever value of r = positive. Interpretation: the two variables move in the same direction. - as X variable goes up, Y increases as well; - as X variable decreases, Y decreases as well. b) Negative correlation: whenever value of r = negative. Interpretation: the two variables move in opposite direction: - as X variable goes up, Y decreases; -as X variable decreases, Y increases. Test of Significance: H 0 : r = 0

18. Table 1. Correlations of Assessment of Socialism in 1988, 1993, 1998, 2003 and 2008 a Correlation coefficients are calculated on panel sample, N = 938; assessment of socialism = 5 categories. **p < 0.01. Year Assessment of Socialism a 19881993199820032008 19881.00 0.150 ** 0.139 ** 0.0620.064 1993 0.150 ** 1.00 0.301 ** 0.310 ** 0.313 ** 1998 0.139 ** 0.301 ** 1.00 0.316 ** 0.318 ** 20030.0620.310 ** 0.316 ** 1.000.381 ** 20080.0640.313 ** 0.318 ** 0.381 ** 1.00

19. Bivariate Regression = used to predict the dependent variable (DV) from the independent variable (IV); -finds the best fitting straight line between X & Y: the line that goes through the means of X and of Y, and minimizes the sum of squared errors (distances) btw. the data points and the line. (Prediction model) predicted value for the dependent variable, Y a = the intercept (the value of Y when X = 0) b = the regression coefficient (the slope), indicating the amount of change in Y given a unit change in X X =the independent variable We call a & b unstandardized coefficients: both are interpreted in the original units (the measurements) of the variables.

20. b= the sum of products of dev. from mean/sum of squares for X

21. Method of Least Squares The prediction errors, called residuals, are defined as the differences between observed and predicted values of Y Y = a + bX + e;  e = Y- (a +b*X)  The regression line minimizes the sum of error terms:

22. Interpretation of b b > 0, positive relationship (X has a positive effect on Y) b < 0, negative relationship (X has a negative effect on Y) b = 0, no relationship (X has no effect on Y) Hypothesis Testing 1) ANOVA and regression 2) T-test for the effect of the independent variable (H0: b = 0)