Non parametric & parametric techniques

Parametric vs non-parametric tests (contrast)

parametric :

• known specifc distribution - ie Normal

• big samples

• specific parameters

• usually quantitative data used

non-para :

• dont know dist

• small sample size

• qualitative data

• dont require specific parameters

• tests rely on ranks , signs , frequencies

• considering relative position

2 types of data (make a tree , whats defining factor of ratio or interval quant data, differences btwn the 2 , give 2 examples )

1. Quantitative - diffrence?- consider if theres a zero point

• Ratio : meaningful differences + a true 0 → Height , weight, temp in K

• Interval: meaningful differences (no true 0) → time, temp *C, IQ score

2. Qualitative

• Nominal : just categories , no order → make of car, gender, a month name

• Ordinal : categories w a order, relative rank → education level (UG,PG,PhD), grade symbol (A,B…)

Non-para test we look at

1. Wilcoxon signed rank sum

2. Mann-Whitneyy-Wilcoxon test

3. Kruskal Wallace

4. Friedmann

5. Spearman’s rankcor

Ranking data

R formula = rank

1. order data low to high (unless otherwise specified)

2. assign ranks according to rel position of data

3. Ties ? - Yes : take avg of ranks

Wilcox signed rank sum test (when we use, what r we testing, null & alt hypothesis, data used,assumptions

when : comparing 2 matched (related in some way) quantitative samples

• what we test : median of differences

Hypotheses

• H0: median of diffs = 0

• H1 : median of diffs /= , >, < 0

Data we look at:

• 2 paired samples

• Quantitative interval OR ratio data

Assumptions:

• under H0 : pop of differences in each grp symmetric around median

• the n paired diffs independent & random samples

Calculating test stat ( whats W + whats it measuring + overall what r we investigating, whats it mean when W near 0 , what does it mean & what do we do when n>10

1. calc diff for each pair

2. ignore pairs w diff = 0

3. n = no. of non-zero diffs

4. sign(±) of paired diffs

5. Sort (small to big) abs val of the diffs

6. Put sign back

7. Calc W (test stat) = sum of signed ranks

Note : W close to 0 means no diff

• wanna see if theres a diff btwn our matched samples

• W is measuring if the signs across the ranks are randomnly distributed or theres a pattern

When sample size >10:

• sampling dist of W normal

• 1. calc test stat z= ( W - miu )/ std dev (sigma) as normal

• 2. do hyp test as normal

• 3. p-value → pnorm(test stat,lower.tail = F )

Mann-Whitney-Wilcoxon test

• objective : see if 2 independent samples of ordinal or quantitative data have same median

• Assumptions :

• 2 random saample sizes n1 & n2

• data quantitative or ordinal

• samples + obs in each sample independent

• only differ in median (if the median is diff)

• H0 : pop medians same

• H1 : pop means diff , 1st median to the right/left of 2nd ie bigger or smaller

test stat

1. combine data set to one

2. rank all obs big → small

3. sum of ranks (note which sample each obs comes from)→ T1 T2

small sample (n1 and or n2 <10:

• T = T1

• Critical region : TU = n1(n1 + n2+1) - TL

• Rejject H0 if : T <= TL or T>= TU

Big sample:

• T = normal dist

• calc z score (test stat) : z= T-miuT/sigmaT miuT= n1(n1 + n2+1)/s sigmaT= root (n1n2(n1 + n2+1)/12)

• Test stat → qnorm(alpha,lower.tail = F / T )

• p value → pnorm(teststat,lower.tail = F)

• Reject H0 if T <= TL or T >= TU (same)

Kruskal - Wallace test

when? - compare 2+ independent grps/samples ordinal or quantitative data by medians (ie see if same pop)

• what we test: medians across groups

• Hypotheses:

  • H0: all population medians are equal

  • H1: at least two population medians different

• Data we look at:

  • ni >= 3 (at least 3 obs in each sample)

  • Quantitative or ordinal data

• Assumptions:

  • treatment lvls and obs within treatment lvls are independent and random

  • distribution of scores in each group is the same shape

Test stat :

1. Combine obs from all grps to one sample

2. Rank small to big

3. Avg the ranks of tied obs

4. Calc sum of ranks T1 → TK

5. compute H :

6. Critical region : H folows chi squared dist , K-1 df , 1 sided upper tail test

7. Reject H0 : H >= p val

8. R : kruskal.test(formula = num ~ name, data = kw1

kruskal.test(formula = num ~ name, data = kw2

Friedmann Test

When? - compare 2+ independent grps/samples ordinal or quantitative data using matched or blocked samples ,by medians (ie see if same pop)

Data

• ordinal or quantitative

• data from blocked experiment w b blocks (similar exp units grped )

• k = no. of treatments

• b = no. of blocks

Assumptions

• measurements within block dependent

• measurements from diff blocks independent

• No interaction btwn blocks + treatments (whats administered to exp units - pops/variables being compared in test)

Friedmann test

• H0: all population medians are equal

• H1: at least two population medians different

1.Rank small to big

2. Avg the ranks of tied obs

3. Calc sum of ranks T1 → TK

4. Test stat ( Fr) : if k/b >= 5 ; chi squared dist K-1 df

5. Reject H0 : Fr >= critical val OR p-value < alpha

Spearmann rank correlation coefficient test

when? - measure association btwn 2 samples/variables of quantitative or ordinal data

Data

• n randomnly chosen paired observations

• n = no. pairs in the data

• n >= 10 → normal dist

Assumptions

• both variables at least ordinal or quabtitative & at least 1 variable not normal

Spearmann test

1. H0 : ps = 0 (no associtaion between the 2 vars in underlying pop)

H1 : ps =/ 0 ( is association) , > 0( correlation is positive) , <0 (correlation is negative)

2. Test stat :

1. rank pops X & Y separately

2. calc difference d within each pair of ranks , d = rank(xi) - rank (yi)

3. (large - normal dist - sample) : z = rs * root(n-1)

4. Reject H0 : p-value <= alpha

5. p- value for norm dist = pnorm()

Advantage of non-para tests - 4 & Disadvange

Advantages

1. useful when assumptions of para test uncertain

2. useful when n small

3. few assumptions

4. not restricted to quantitative data

Disadvantage

1. Info gets lost thru ranking & taking signs → less power compared to parametric tests