Taehyun Jung taehyun.jung@circle.lu.se CIRCLE, Lund University

1. 10:30-12:00 December 10 2012 For Survey of Quantitative Research, NORSI Session 2: Basic techniques for innovation data analysis. Part I: Statistical inferences and comparisons of groups Taehyun Jung taehyun.jung@circle.lu.se CIRCLE, Lund University

2. Objectives of this session

3. Contents • Correlation • Statistical Inference and Hypothesis Testing • t-Test • Confidence Interval • Chi-square Statistic

4. Correlation

5. A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. • As in any graph of data, look for the overall pattern and for striking departures from that pattern. • Form: linear, curved, clusters, no pattern • Direction: positive, negative, no direction • Strength: how closely the points fit the “form” • An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship. Scatterplot

6. No relationship Nonlinear Linear

7. The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form. • With a weak relationship, for any x you might get a wide range of y values. • With a strong relationship, you can get a pretty good estimate of y if you know x.

8. correlation The sample Pearson’s correlation coefficient r measures the strength of the linear relationship between two quantitative variables. The correlation coefficient rmeasures the strength of the linear relationship between two quantitative variables. • r is always a number between -1 and 1. • r > 0 indicates a positive association. • r < 0 indicates a negative association. • Values of r near 0 indicate a very weak linear relationship. • The strength of the linear relationship increases as r moves away from 0 toward -1 or 1. • The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship. • Part of the calculation involves finding z, the standardized score • Allows us to compare correlations between data sets where variables are measured in different units or when variables are different

9. Correlation makes no distinction between explanatory and response variables. r has no units and does not change when we change the units of measurement of x, y, or both. Positive r indicates positive association between the variables, and negative r indicates negative association. The correlation r is always a number between -1 and 1. Cautions • Correlation requires that both variables be quantitative. • Correlation does not describe curved relationships between variables, no matter how strong the relationship is. • Correlation is not resistant. r is strongly affected by a few outlying observations. • Correlation is not a complete summary of two-variable data. Facts About Correlation

10. Strength: How closely the points follow a straight line. Direction is positive when individuals with higher x values tend to have higher values of y “r” ranges from −1 to +1

11. Correlations are calculated using means and standard deviations and thus are NOT resistant to outliers. Just moving one point away from the general trend here decreases the correlation from −0.91 to −0.75. Influential points

12. Statistical Inference and Hypothesis Testing

13. The normal distribution has the bell-shaped (Gaussian) form. • arises from the central limit theorem, which states that under mild conditions, the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution • very tractable analytically Normal distribution

14. Assumption: • Null hypothesis : • Alternative hypothesis: • We will suppose that we have observations on a random variable with a normal distribution with unknown mean m and that we wish to test the hypothesis that the mean is equal to some specific value . Testing a hypothesis relating to the population mean

15. Suppose that we have a sample of data for the example model and the sample mean is . Would this be evidence against the null hypothesis ? • No, it is not. It is lower than , but we would not expect to be exactly equal to because the sample mean has a random component. • If the null hypothesis is true, the probability of the sample mean being one standard deviation or more above or below the population mean is 31.7%.

16. four standard deviations above the hypothetical mean? • the chance of getting such an extreme estimate is only 0.006%. • We would reject the null hypothesis The usual procedure for making decisions is to reject the null hypothesis if it implies that the probability of getting such an extreme sample mean is less than some (small) probability p. • For example, the probability of getting such an extreme sample mean is less than 0.05 (5%) • The 2.5% tails of a normal distribution always begin 1.96 standard deviations from its mean

17. Decision rule (5% significance level): Reject • (1) if > + 1.96 s.d.or (2) if < – 1.96 s.d. • (1) if >1.96 or (2) if < 1.96 • Type I error: rejection of H0 when it is in fact true. • Probability of Type I error: in this case, 5% • Significance level (size) of the test is 5%.

18. We can of course reduce the risk of making a Type I error by reducing the size of the rejection region.

19. t-Test

20. What if we do not know the standard deviation? The test statistic has a t distribution instead of a normal distribution s.d. of X known s.d. of Xnot known discrepancy between hypothetical value and sample estimate, in terms of s.d.: discrepancy between hypothetical value and sample estimate, in terms of standard error (s.e.): 5% significance test: reject H0: m = m0 if z > 1.96 or z < –1.96 5% significance test: reject H0: m = m0 if t > tcrit or t < –tcrit

21. For a sample of size n, the sample standard deviation s is: • n − 1 is the “degrees of freedom.” • The value s/√n is called the standard error of the mean SEM. • Scientists often present their sample results as the mean ± SEM.

22. When the number of degrees of freedom is large, the t distribution looks very much like a normal distribution (and as the number increases, it converges on one) Then, why t-dist? • Although the distributions are generally quite similar, the t distribution has longer tails than the normal distribution, the difference being the greater, the smaller the number of degrees of freedom • the rejection regions have to start more standard deviations away from zero for a t distribution than for a normal distribution

23. A certain city abolishes its local sales tax on consumer expenditure. A survey of 20 households shows that, in the following month, mean household expenditure increased by $160 and the standard error of the increase was $60. We wish to determine whether the abolition of the tax had a significant effect on household expenditure. • We take as our null hypothesis that there was no effect: • The test statistic is • The critical values of t with 19 degrees of freedom are 2.09 at the 5 percent significance level and 2.86 at the 1 percent level. • Hence we reject the null hypothesis of no effect at the 5 percent level but not at the 1 percent level. Example

24. The t tests are exactly correct when the population is distributed exactly normally. However, most real data are not exactly normal. The t tests are robust to small deviations from normality. This means that the results will not be affected too much. Factors that do strongly matter are: • Random sampling. The sample must be an SRS from the population. • Outliers and skewness. They strongly influence the mean and therefore the t procedures. However, their impact diminishes as the sample size gets larger because of the Central Limit Theorem. • Specifically: • When n < 15, the data must be close to normal and without outliers. • When 15 > n > 40, mild skewness is acceptable, but not outliers. • When n > 40, the t statistic will be valid even with strong skewness. Robustness

25. Confidence interval

26. Any hypothesis lying in the interval from mmin to mmax would be compatible with the sample estimate (not be rejected by it). We call this interval the 95% confidence interval. Confidence interval (2) (1) X mmin–1.96sd mmin–sd mmin mmin+sd mmax–sd mmax mmax+sd mmax+1.96sd

27. Standard deviation known 95% confidence interval – 1.96 sd< m < + 1.96 sd 99% confidence interval – 2.58 sd< m < + 2.58 sd Standard deviation estimated by standard error 95% confidence interval –tcrit (5%) se< m < + tcrit (5%) se 99% confidence interval –tcrit (1%) se< m < + tcrit (1%) se

28. Chi-square statistic

29. STATA command • . tab dused_ndef largef, col chi • Pearson chi2(1) = 11.4978 Pr = 0.001 Can we conclude that large firms use patents more strategically than small firms based on this table? Use of patents by firm size

30. The chi-square statistic () measures how far the sample is from what we “expect” to see in a random sample from a population with NO relationship. • If is too far from what we expected, we conclude that the sample did not come from a population with no relationship and therefore conclude that the variables must be related in the population. • H0: There is no relationship between categorical variable A and categorical variable B. • Ha: There is some relationship between categorical variable A and categorical variable B. This alternative hypothesis is not really one-sided (> or <) or two-sided (). It can be called “many-sided” because it allows any kind of relationship between variables A and B to count. Chi-square hypothesis test

31. We want to test the hypothesis that there is no relationship between these two categorical variables (H0). • To test this hypothesis, we compare actual counts from the sample data with expected counts given the null hypothesis of no relationship. • The expected count in any cell of a two-way table when H0is true is: The chi-square statistic (c2) is a measure of how much the observed cell counts in a two-way table diverge from the expected cell counts.

32. Large values for represent strong deviations from the expected distribution under the H0, and provide evidence against H0. However, since is a sum, how large a is required for statistical significance will depend on the number of comparisons made.

33. For the chi-square test, H0states that there is no association between the row and column variables in a two-way table. The alternative is that these variables are related. If H0is true, the chi-square test has approximately a χ2 distribution with (r − 1)(c − 1) degrees of freedom. The P-value for the chi-square test is the area to the right of c2 : P(χ2 ≥ X2).

34. Probability of rejecting the null hypothesis if H0 is true • Typically, .05 or .01 significance level • With a significance level of .05 and 1 df, X2=3.84; we will reject H0 when X2* is greater than 3.84 and accept H0 when X2* is less than 3.84. • if the null hypothesis is true (if the variables are not related in the population), we will still (incorrectly) reject H0 (conclude that the variables are related in the population) about 5 times (or 1 time) in 100 hypothesis tests A key step in the hypothesis test is deciding how willing we are to make a Type I error. (We must take some chance of rejecting a true null hypothesis or we will have no chance of rejecting a false one.) • Type I error: incorrectly rejecting the null hypothesis. • Type II error: Incorrectly accepting the null hypothesis. Significance Level (alpha)