Basic statistics

from scipy import stats 
import seaborn as sns
import matplotlib.pyplot as plt

T-test

두 집단간의 평균 차이가 유의미한지 확인할 때 사용.

  1. 두 집단은 독립적일 것.
    • 독립이 아니면 Pared t-test.
  2. 데이터가 정규분포를 따르는가.
    • 시각적 : Q-Q plot
    • 정량적 : Shapiro-Wilk normality test. H0=데이터가 정규분포를 따른다. alpha=0.05.
    • 정규분포를 따르지 않으면 Wilcoxon test.
  3. 집단이 둘인지.
    • 두 집단을 넘으면 ANOVA 시행.
  • T-score = (mA-mB)/[SAB(1/nA + 1/nB)] :
  • SAB = sqrt([(nA-1)SA^2 + (nB-1)SB^2]/(nA+nB-2))
# T-test
stats.ttest_ind(group1, group2, equal_var=False) # Default equal_var=True
# 분산이 같다고 가정할지, 아닐지 결정

# Shapiro-Wilk test
stats.shapiro(group1)


fig=sns.distplot(group1, kde=False, fit=stats.norm, label='G1')
fig=sns.distplot(group2, kde=False, fit=stats.norm, label='G2')
fig.lines[0].set_linestyle(":")
plt.title('G1: '+str(stats.shapiro(group1)[1])+', G2: '+str(stats.shapiro(group2)[1]))
plt.grid(ls='--',alpha=0.4)
plt.legend()
plt.savefig('dist',dpi=300)

kolmogorov-Smirnov test

연속형 데이터가 정규분포를 따르는지 확인할때 쓰는 검정방법.

Statistical error

  • Twenty Statistical Errors Even YOU Can Find in Biomedical Research Articles, Tom Lang, CMD 45(4):361-370, 2004.

  • Reporting measurements with unnecessary precision.

    • the smallest P value that need be reported is P < 0.001.
  • Dividing continuous data into ordinal categories without explaining who or how.
    • For exmaple, Height cm to ‘short, normal, and tall’. why you choose, how the boundaries are determined.
  • Reporting group means for paired data without reporting within-pair changes.
  • Using descriptive statistics incorrectly.
    • In markedly non-normal distributions, the mean and standard deviation do not communicate the shape of the distribution well. Instead, the median is recommended.
  • Using the standard error of the mean as a descriptive statistic or as a measure of precision for an estimate.
    • the mean and standard deviation are the preferred summary statistics for (normally distributed) data, and the mean and 95% confidence interval are preferred for reporting an estimate and its measure of precision. (95% CI = 2 SEM)
    • e.g. “The mean value was 72kg (95% CI = 70.4 to 73.6kg).
  • Reporting only P values for results.
    • For main results, report the absolute difference btw groups and the 95% confidence interval for the difference.
  • Not confirming that the data met the assumptions of the statistical tests used to analyze them.
    • Tests may not give accurate results if their assumptions are not met.
  • Using linear regression analysis without establishing that the relationship is, in fact, linear.
  • Not accounting for all data and all patients.
    • Missing data is a common but irritating reporting problem made worse by the thought that the author is careless, lazy, or both.
  • Not reporting whether or how adjustments were made for multiple hypothesis tests.
    • Multiple testing is often desirable, and exploratory analyses should be reported as exploratory.
  • Unnecessarily reporting baseline statistical comparisons in randomized trials.
    • In some cases, the P-value do not need to be reported.

SEM

: Standard Error of the Mean

  • Misuse of standard error of the mean (SEM) when reporting variability of a sample, P. Nagele, British J. of Anaesthesia 90(4):514-16, 2001.

  • Standard deviation (SD)

    • the variability between individuals in a sample.

  • Standard error of the mean (SEM)

    • the uncertainty of how the sample mean represents the population mean.
    • used in inferential statistics to give an estimate of how the mean of the sample is related to the mean of the underlying population.

  • The SD tells us the distribution of individual data points around the mean, and the SEM informs us how precise our estimate of the mean is.

  • In general, the use of the SEM should be limited to inferential statistics where the author explicitly wants to inform the reader about the precision of the study, and how well the sample truly represents the entire population. Thus, in inferential statistics, the use of SEM is valid but the CI is more valuable.

Error and residuals

  • A statistical error is the difference between an observed value in sample and the statistical unit in population. For instance, the mean height of population is 1.75m. When the randomly chosen man is 1.70m tall, then the "error" is -0.05m.

  • A residual is the difference between an observed value in sample and the statistical unit in sample.

P-value

  • A p-value is the probability that 1) random chance generated the data, or 2) something else that is equal or 3) rarer.