CHI SQUARE TEST
CHI-SQUARE GOODNESS-OF-FIT TEST (only one category)
CHI-SQUARE TEST OF INDEPENDENCE (two categories)
Condition:
1. Use only absolute frequencies (not percentage) in the calculation.
2. Make sure that each individual is included in only one category
3. Check that all expected frequencies are at least 5.
When the data is about small sample or there is no homogeneity of variance between groups.
1. CHI SQUARE
condition: one sample one test
X2(2)=20.6, p<0.05
1. df=r-1
2. p value
3. two tailed test only
p.188-189
1. BIVARIATE CORRELATION
To test correlation ship between two variables
2. REGRESSION EQUATION
Line of best fit and Standard error of estimates
Y'=bX+a
Multiple regression: when you determine to include multiple independent variables, be sure of the correlation between each independent variable and dependent variable, and the independence between every pair of independent variables.
I. INDEPENDENT SAMPLE T TEST
Independent-samples t test assumes the distributions of a (or more) variables in two groups is same. (untold assumption: the internal nature of the group is the independent variable)
condition: two indepedent samples take one test only.
t(58)=-1,14, p>0.05
1. df (degree of freedom)=n1+n2-2
2. p value
3. one-tailed or two-tailed test
Effect size (ES) is a measure of the strength of the relationship between two variables in a statistical population, or a sample-based estimate of that quantity. An effect size calculated from data is a descriptive statistic that conveys the estimated magnitude of a relationship without making any statement about whether the apparent relationship in the data reflects a true relationship in the population. In that way, effect sizes complement inferential statistics such as p-values. In certain sense, that means p-value tell you the probability to take the wrong decision to reject or accept the null hypothesis, while effect size or correlation coefficient tells you strength of the relationship between variables.
Cohen's d: (X1-X2)/SD
http://www.uccs.edu/~faculty/lbecker/
II. ANOVA (SIMPLE ANALYSIS OF VARIANCE / ONE WAY ANALYSIS OF VARIANCE)
One-way analysis of variance assumes the distribution of a (or more) variables in more than two groups is same.
condition: more than two groups take one test (one variable).
F(2,27)=8.80, p<0.05
1. df (between)=k-1; df (within)=N-k; df(total)=N-1
2. p value
3. only two tailed test
To determine where is the difference, run post-hoc test with Bonferroni analysis
III. FACTORIAL ANALYSIS OF VARIANCE
Factorial analysis of variance assumes the distribution of a (or more) variables in more than two groups categorized by two variables is same.
Main effect and interaction effect
SPSS use univariate analysis of variance, for it concerns only one explained variable or one dependent variable.
*if more than two variables, Holy Grail Analysis of Variance
IV. PARED SAMPLE T TEST
Pared sample t test assumes the distribution of a (or more) variable in one group before and after an experiment is same. (untold assumption: the experiment is the independent variable)
condition: only one group takes two tests.
t(24)=2.45, p<0.05
1. df=n-1
2. p value
3. one tailed or two tailed test.
INFERENTIAL STATISTICS
Inferential statistics is used to infer the information about population from the observation of a sample.
Population-Sampling frame-Sampling pool (probability/random sampling or non-probability sampling)-Sample
Estimation
Estimation of numerical data
The principal is that if we can draw all samples from a population, then the distribution of sample means will take a normal curve with the sample means that equal the population means situated in the center. However, we can only have the observation of one sample, and we don't know where exactly located that sample in the distribution of samples, in other words, we don't know whether we have a typical representative sample or not. We then estimate the standard deviation of population from the sd of sample by divided (n-1). That is also why the standard deviation calculation is often divide variance by (n-1) not n. We then have standard deviation of sampling distribution of means by divide sd of population by square root of sample size. According to nature of the normal distribution, we estimate the range of population means from the means of sample with a certain confidence level. Say (sample means-2sd; sample means+2sd) with 95% of confidence level.
Student t shows the adjustment of confidence level when the sample size varies. Basically when the sample size is higher than 30, it respects the normal distribution.
Estimation of categorical data
The principal is the same only we treat category percentage here. The calculation of standard deviation of sampling distribution of percentage will be:
Another difference is that the confidence level of the estimation is not only influence by the sample size but also by the category percentage in the population. As there is no student t table for categorical data, when have another benchmark: smaller category percentage of sample * sample size > 1000, if not increase sample size.
Hypothesis testing
A good hypothesis should:
1) reflect the theoretical background and available references
2) short and clear affirmative sentence
3) relationship between variables
4) testable
Inferencial statistics basic concept:
1. Null hypothesis vs. research hypothesis (nondirectional and two tailed test; directional and one tailed test)
Null hypothesis means in the whole population, there is no difference between two samples, or in other words the independent variables doesn't cause the different distribution of dependent variable. the observed variance is due to the coincidence.
2. Normal Curve
Mean=Median; 34.13% of the data is between mean and mean+1 standard deviation (or -1); 13.59% between +1 and +2 sd; 2.15% between +2 and +3 sd, 0.13% more than +3 sd. (Probability: < +1 84%; (+1,+2) 14%; (+2, +3) 2.15%)
3. Standard score (Z)
measure the distance tween a data x and the sd, which can infer the probability of the appearance of a data x.
4. Significance rate: 5%
Z score is to determine whether one event is caused by purely chance or just a result of casual distribution of probability. thus, if one event is hardly to happen under normal condition, that means below 5% (Z>I1.65I), then null hypothesis is wrong.
5. Significance level (p)
6. Degree of freedom (df)
Type I error (α): while null hypothesis true you decide null hypothesis is wrong. and Type II error (β): vice versa
null hypothesis is about the total and can not be verified directly.
Type II error is related to the sample size
I think the most valuable part of today's chapiter is about the design of inferencial statistics. the significance of statistic itself is meaningless. the most important task goes to the analytic work generating the hypothesis about variables. in the practical aspect, the sample construction is also more important then the verification itself.
How to control the variables and how to choose the appropriate instruments is what I'll continue to learn. but I don't think it will be more difficult and more complicated than the original academic analysis, especially with help of computer software.
P.108
DESCRIPTIVE STATISTICS
Statistics is the main quantitative tool used by all social scientists, for this reason I make this series of posts to record my self study outcomes.
Revision of general concepts:
Type of data: Categorical variable (nominal, ordinal); Numerical variable (interval). ARRAY
Mode and Variation ratio for nominal variables; Median and Midspread for ordinal variables; Mean and SD or Median and midspread for interval variables.
Percentage, proportion, ratio and rates of occurrence
Importance of Z score, for different types of data, the main idea should be keep as many information as possible that the original data contain. Thus, for example when we compare the advance that China achieved in GDP and Human development, we should avoid reduce the numerical data to ordinal one, but use standardized Z score to compare the historical development in these two area.
Index construct: Index-Dimensions-Indicators-Variables-Values. To combine different values calculated in different units, we add up its respective rank standings by simply add up or weighted sum. We then have index rankings through index scores.
Central tendency- Mean(arithmetic mean), Weighted mean, Median, Mode
Variability- Standard deviation, Variance, Range, Midspread (interquartile range: upper quartile-lower quartile); Variation ratio
Skewness and Kurtosis: negative skewness when mean is larger than median, positive when mean is smaller than median. SK=3(mean-median)/sd
CORRELATION
Direction (positive or negative); Nature (Forms of line); Strength (How well to predict a dependent variable from knowing independent variable)
Correlation efficient = (Original error - Remaining error)/Original error
Correlation between categorical variables.
Cross tabulation with column percentage while independent variables in column; Mosaic plot graph.
Lambda correlation coefficient: ((Total-total mode)-((Category 1_total-Cateogry 1_mode)+(Category 2_total-Category 2_mode)))/(Total-total mode)).
When the category mode numbers are in the same row, then lambda correlation coefficient is always a zero.
Correlation between numerical variables.
Scatter graph with best-fit line (regression line)
Y=aX+b, which must pass the point (X,Y) when X=mean of DV and Y=mean of IV.
Coefficient of determination = (Variance of DV-Variance of errors)/(Variance of DV). Error is the difference between observed value and predicted value.
Correlation between categorical and numerical variables.
eta squared coefficient = ((Variance of DV-((Variance of category 1)+(Variance of category 2))/2)/(Variance of DV)
Coefficient of determination is the square of correlation coefficient (pearson r) to demonstrate the percentage of variance of X can be explained by variance of Y. Thus when X and Y is correlated, the variance of X and Y may both explained partly by a common factor Z. There's no implied causality between X and Y.
Choose the proper coefficient according to type of data set.
Pearson correlation score for numerical data
Chi-square for categorical data
Spearman rank correlation coefficient for ordinal variables
Point-biserial correlation coefficient for a categorical variable and a categorical variable
Rank-biserial correlation coefficient for a categorical variable and a ordinal variable
In my paper, cross tabulation with column percentage could be used to describe the difference of distribution of projects between public and private enterprises, while scatter graph could be used to show the impacts of GDP, Natural resources and Institutions on total number of projects.
-- 发送自我的 iPad
