ANOVA

Single Factor ANOVA

A number of yarns have been spun from wool/rayon blend, with different proportion of the two components. 6 different variants have been manufactured, and for each variant a total of 8 measurements of the yarn strength have been made in [cN]. The results have been summarized in Table.

Determine whether the change in the proportion of the components has an influence on the yarn strength.

Proportion of components
40/60	45/55	50/50	55/45	60/40	65/35
100	93	101	127	111	69
141	120	87	83	84	86
147	123	88	80	83	80
126	105	105	87	96	82
133	109	93	96	83	80
101	103	102	109	90	99
148	130	73	68	101	81
128	143	94	97	64	71

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

df = pd.DataFrame({
    'proportion': ['40/60', '40/60', '40/60', '40/60', '40/60', '40/60', '40/60', '40/60', '45/55', '45/55', '45/55', '45/55', '45/55', '45/55', '45/55', '45/55', '50/50', '50/50', '50/50', '50/50', '50/50', '50/50', '50/50', '50/50', '55/45', '55/45', '55/45', '55/45', '55/45', '55/45', '55/45', '55/45', '60/40', '60/40', '60/40', '60/40', '60/40', '60/40', '60/40', '60/40', '65/35', '65/35', '65/35', '65/35', '65/35', '65/35', '65/35', '65/35'],
    'strength': [100, 141, 147, 126, 133, 101, 148, 128, 93, 120, 123, 105, 109, 103, 130, 143, 101, 87, 88, 105, 93, 102, 73, 94, 127, 83, 80, 87, 96, 109, 68, 97, 111, 84, 83, 96, 83, 90, 101, 64, 69, 86, 80, 82, 80, 99, 81, 71]})


model = ols('strength ~ proportion', data=df).fit()
sm.stats.anova_lm(model, typ=1)

	df	sum_sq	mean_sq	F	PR(>F)
proportion	5.0	12869.75	2573.950000	11.449167	5.219617e-07
Residual	42.0	9442.25	224.815476	NaN	NaN

Two-Factor ANOVA Without Replication

А sweater is produced on three different machines. The table below shows the number of defective sweaters produced in the last six months. Тhe table also shows that every month a different operator works with the machines. Determine whether the different number of defective sweaters is due to the type of machine or the operator.

	Machine 1	Machine 2	Machine 3
Operator 1	75	75	90
Operator 2	70	70	70
Operator 3	50	55	75
Operator 4	65	60	85
Operator 5	80	65	80
Operator 6	65	65	65

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

df = pd.DataFrame({
    'operator': ['Operator 1', 'Operator 2', 'Operator 3', 'Operator 4', 'Operator 5', 'Operator 6', 'Operator 1', 'Operator 2', 'Operator 3', 'Operator 4', 'Operator 5', 'Operator 6', 'Operator 1', 'Operator 2', 'Operator 3', 'Operator 4', 'Operator 5', 'Operator 6'],
    'machine': ['Machine 1', 'Machine 1', 'Machine 1', 'Machine 1', 'Machine 1', 'Machine 1', 'Machine 2', 'Machine 2', 'Machine 2', 'Machine 2', 'Machine 2', 'Machine 2', 'Machine 3', 'Machine 3', 'Machine 3', 'Machine 3', 'Machine 3', 'Machine 3'],
    'defectives': [75, 70, 50, 65, 80, 65, 75, 70, 55, 60, 65, 65, 90, 70, 75, 85, 80, 65]})


model = ols('defectives ~ operator + machine', data=df).fit()
sm.stats.anova_lm(model, typ=1)

	df	sum_sq	mean_sq	F	PR(>F)
operator	5.0	750.0	150.0	3.157895	0.057399
machine	2.0	525.0	262.5	5.526316	0.024181
Residual	10.0	475.0	47.5	NaN	NaN

Two-Factor ANOVA With Replication

Study the influence of the needle number and the density of the seam on the strength of the seam in [N]. The study has been conducted with three different needle numbers: 70, 80 and 90. the density of the seam being 3, 4 and 5 stitches/cm. 5 tests have been conducted for each combination of the factors. The data from the measurements is given in the following table:

Density 3	70 Needles	80 Needles	90 Needles	Density 4	70 Needles	80 Needles	90 Needles	Density 5	70 Needles	80 Needles	90 Needles
	192.9	155.3	183.5		243.6	230.1	252.7		267.7	218.1	229.7
	220.9	170.3	188.5		254	201.5	210		277.2	250.4	253.2
	199.1	168.2	184.5		261.1	224.9	218		264.4	220.4	203.5
	195.6	160.5	174.5		255.7	230.1	245.7		293.3	156	247.1
	176	176.6	185.2		224.1	209.9	233.4		256.3	218.3	271.6

import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols

df = pd.DataFrame({
    'density': ['Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 3', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4', 'Density 4','Density 4', 'Density 4', 'Density 4', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5', 'Density 5'],
    'needles': ['70 Needles', '70 Needles', '70 Needles', '70 Needles', '70 Needles', '80 Needles', '80 Needles', '80 Needles', '80 Needles', '80 Needles', '90 Needles', '90 Needles', '90 Needles', '90 Needles', '90 Needles', '70 Needles', '70 Needles', '70 Needles', '70 Needles', '70 Needles', '80 Needles', '80 Needles', '80 Needles', '80 Needles', '80 Needles', '90 Needles', '90 Needles', '90 Needles', '90 Needles', '90 Needles', '70 Needles', '70 Needles', '70 Needles', '70 Needles', '70 Needles', '80 Needles', '80 Needles', '80 Needles', '80 Needles', '80 Needles', '90 Needles', '90 Needles', '90 Needles', '90 Needles', '90 Needles'],
    'strength': [192.9, 220.9, 199.1, 195.6, 176, 155.3, 170.3, 168.2, 160.5, 176.6, 183.5, 188.5, 184.5, 174.5, 185.2, 243.6, 254, 261.1, 255.7, 224.1, 230.1, 201.5, 224.9, 230.1, 209.9, 252.7, 210, 218, 245.7, 233.4, 267.7, 277.2, 264.4, 293.3, 256.3, 218.1, 250.4, 220.4, 156, 218.3, 229.7, 253.2, 203.5, 247.1, 271.6]})


model = ols('strength ~ density + needles + density:needles', data=df).fit()
sm.stats.anova_lm(model, typ=2)

	sum_sq	df	F	PR(>F)
density	31157.852444	2.0	44.990762	1.614067e-10
needles	11655.701778	2.0	16.830393	6.912805e-06
density:needles	1486.102222	4.0	1.072938	3.841812e-01
Residual	12465.700000	36.0	NaN	NaN