Two-Way ANOVA

1 Introduction
2 Hypthoses
3 ANOVA Table
4 Degrees of Freedom
5 Critical Values
6 Find Sum of Squares and Mean Squares
7 F-Test
8 ANOVA Table
9 Do It In R
10 F-Distribution Table \(\alpha\) = 0.05

library(data.table)

1 Introduction

So this is an overly pedantic explanation of two-way ANOVA that describes how to do the math both by hand and using R. This will use the CO2 dataset which shows the amount of co2 absorbed by the same species of grass that originated from either Mississippi or Quebec and were measured after being either chilled overnight or unchilled.

co2 <- data.table(CO2)
head(co2)

##    Plant   Type  Treatment conc uptake
## 1:   Qn1 Quebec nonchilled   95   16.0
## 2:   Qn1 Quebec nonchilled  175   30.4
## 3:   Qn1 Quebec nonchilled  250   34.8
## 4:   Qn1 Quebec nonchilled  350   37.2
## 5:   Qn1 Quebec nonchilled  500   35.3
## 6:   Qn1 Quebec nonchilled  675   39.2

The dataset measures co2 uptake at different levels of ambient co2 concentrations so I’ll just average those for each plant.

co2[,.(meanUptake = mean(uptake)), by = .(Plant,Type,Treatment)]

##     Plant        Type  Treatment meanUptake
##  1:   Qn1      Quebec nonchilled   33.22857
##  2:   Qn2      Quebec nonchilled   35.15714
##  3:   Qn3      Quebec nonchilled   37.61429
##  4:   Qc1      Quebec    chilled   29.97143
##  5:   Qc2      Quebec    chilled   32.70000
##  6:   Qc3      Quebec    chilled   32.58571
##  7:   Mn1 Mississippi nonchilled   26.40000
##  8:   Mn2 Mississippi nonchilled   27.34286
##  9:   Mn3 Mississippi nonchilled   24.11429
## 10:   Mc1 Mississippi    chilled   18.00000
## 11:   Mc2 Mississippi    chilled   12.14286
## 12:   Mc3 Mississippi    chilled   17.30000

So there are three plants that received each treatment and come from each location for a total of 12 plants.

The table looks like this:

	Mississippi	Quebec
Chilled	18, 12.143, 17.3	29.971, 32.7, 32.586
Nonchilled	26.4, 27.343, 24.114	33.229, 35.157, 37.614

2 Hypthoses

Now we state our hypotheses. There are three sets of hypotheses:

\(H_0: \text{Temperature (treatment) doesn't affect co2 uptake}\)

\(H_1: \text{Temperature (treatment) does affect co2 uptake}\)

\(H_0: \text{Plant origin (type) doesn't affect co2 uptake}\)

\(H_1: \text{Plant origin (type) does affect co2 uptake}\)

\(H_0: \text{Plant origin (type) and temperature (treatment) don't affect co2 uptake}\)

\(H_1: \text{Plant origin (type) and temperature (treatment) affect co2 uptake}\)

3 ANOVA Table

We have to fill in this ANOVA table to find out if the F-test value is greater or less than the C.V.

	d.f.	SS	MS	F
Treatment (A)	a-1	\(SS_A\)	\(SS_A/(d.f.A)\)	\(MS_A/MS_{Within}\)
Type (B)	b-1	\(SS_B\)	\(SS_{B}/(d.f.B)\)	\(MS_B/MS_{Within}\)
Interaction(AxB)	(a-1)(b-1)	\(SS_{AxB}\)	\(SS_{(AxB)}/(d.f.AxB)\)	\(MS_{(AxB)}/MS_{Within}\)
Within (error)	ab(n-1)	\(SS_{Within}\)	\(SS_W/d.f.W\)
Total	abr-1	\(SS_{Total}\)

4 Degrees of Freedom

To find the critical values we use the F-Distribution table which requires that we have the following:

The Degrees of Freedom for each factor:

\[\begin{align} \text{Factor Treatment: } d.f.N & = (a - 1) = (2-1) = 1 \ & \\ \text{Factor Type: } d.f.N & = (b - 1) = (2-1) = 1 \ & \\ \text{Interaction (A x B) : } d.f.N & = (a - 1)(b - 1) = (2-1)(2-1) = 1 \\ & \\ \text{and the within } & \text{group Degrees of Freedom: }\\ & \\ \text{Within (error): } d.f.D & = ab(n-1) = 2 \cdot 2(3-1) = 8 \end{align}\]

Where:

\(a\) is the number of treatment categories (chilled and nonchilled).
\(b\) is the number of type categories (Quebec and Mississippi).
\(n\) is the number of data values in each group.

Also:

d.f.N stands for Degrees of Freedom Numerator
d.f.D stands for Degrees of Freedom Denominator

5 Critical Values

Based on those numbers and the F-Distribution Table the critical values are:

\(CV_{treatment} = 5.3177\)
\(CV_{type} = 5.3177\)
\(CV_{type \times treatment} = 5.3177\)

6 Find Sum of Squares and Mean Squares

First we have to find the means for each group, then the means of the rows and columns.

	Mississippi	Quebec	Row Mean
Chilled	18, 12.143, 17.3 (Mean = 15.814)	29.971, 32.7, 32.586 (Mean = 31.752)	23.783
Nonchilled	26.4, 27.343, 24.114 (Mean = 25.952)	33.229, 35.157, 37.614 (Mean = 35.333)	30.643
Column Mean	20.883	33.543	27.213

In the following equations:

r = replicants, or the number of values in each group
b = the number of type categories (Quebec and Mississippi)
a = the number of treatment categories (chilled and nonchilled)
\(\bar{X}_{GM}\) = Grand Mean, the mean of all values in the table

6.1 Sum and Mean Squares A

\[\begin{align} SS_{A} = SS_{treatment} & = rb \sum_{i=1}^A (\bar{X}_{i } - \bar{X}_{GM})^2 \\ & & \\ SS_{treatment} & = 3 \times 2 \times[(23.783 - 27.213)^2 + (30.642 - 27.213)^2] = 141.138 \\ & & \\ \text{We're just taking each row mean,} & \text{ subtracting it from the grand mean, squaring each of them, and then adding the result together.} \ & & \\ \text{Then we multiply that result by t} & \text{he number of samples in each group and by the number of }\textbf{ type}\text{ categories.} \\ & & \\ MS_{treatment} & = SS_{treatment} \mathbin{/} df_{treatment} = SS_{treatment} \mathbin{/} (a-1) \\ & & \\ MS_{treatment} & = 141.138 \mathbin{/} 1 = 141.138 \end{align}\]

6.2 Sum and Mean Squares B

\[\begin{align} SS_{B} = SS_{type} & = ra \sum_{j=1}^B (\bar{X}_{j} - \bar{X}_{GM})^2 \\ & & \\ SS_{type} & = 3 \times 2 \times[(20.883 - 27.213)^2 + (33.543 - 27.213)^2] = 480.827 \\ & & \\ \text{Again, pretty much the same as} & \text{ above, we're just taking each column mean, subtracting from the row mean, square the result and} \ & & \\ \text{add the two together. Then mul} & \text{tiply that result by the number of values in each group and the number of} \textbf{ treatment}\text{ categories.} \\ & & \\ MS_{type} & = SS_{type} \mathbin{/} df_{type} = SS_{type} \mathbin{/} (b-1) \\ & & \\ MS_{type} & = 480.827 \mathbin{/} (2-1) = 480.827 \end{align}\]

6.3 Sum and Mean Squares Interaction

\[\begin{align} SS_{(A \times B)} & = r \sum_{j=1}^b \sum_{i=1}^a (\bar{X}_{ij} - \bar{X}_{i} - \bar{X}_{j} + \bar{X}_{GM})^2 \\ & & \\ SS_{(A \times B)} & = 3 \times [(15.814 - 20.883 - 23.783 + 27.213)^2 + (31.752 - 33.543 - 23.783 + 27.213)^2 + \ & & \\ & \hspace{1cm} (25.952 - 20.883 - 30.643 + 27.213)^2 + (35.333 - 33.543 - 30.643 + 27.213)^2] \\ & & \\ SS_{(A \times B)} & = 32.246 \\ & & \\ \text{So in this case we're doing the} & \text{ sum of the square of the group means minus the row and column means plus the grand mean.} \\ & & \\ MS_{(A \times B)} & = SS_{(A \times B)} \mathbin{/} df_{(A \times B)} \\ & & \\ MS_{(A \times B)} & = 32.246 \mathbin{/} = 32.246 \end{align}\]

6.4 Sum and Mean Square Within

\[\begin{align} SS_W & = \sum_{k=1}^r \sum_{j=1}^B \sum_{i=1}^A (X_{ijk} - \bar{X}_{ij})^2 \\ & & \\ SS_W & = (18 - 15.814)^2 + (12.143 - 15.814)^2 + (17.3 - 15.814)^2 + (29.971 - 31.752)^2 + \ & & \\ & \hspace{.5cm} _{\cdot \space \cdot \space \cdot \space \cdot \space \cdot} \ & & \\ & \hspace{.5cm} + (37.614 - 35.333)^2 \\ & & \\ SS_W & = 40.404 \\ & & \\ \text{Here were taking each} & \text{ value in each group, subtracting from the mean of the group and squaring the result.} \\ & & \\ MS_W & = SS_W \mathbin{/} df_{within} = SS_W \mathbin{/} ab(n-1) \\ & & \\ MS_W & = 40.404 \mathbin{/} 8 = 5.050 \end{align}\]

7 F-Test

\[\begin{align} F_A = & \frac{MS_A}{MS_W} = \frac{141.138}{5.050} = 27.948 \\ & & \\ F_B = & \frac{MS_B}{MS_W} =\frac{480.827}{5.050} = 95.213 \\ & & \\ F_{(A \times B)} = & \frac{MS_{(A \times B)}}{MS_W} = \frac{32.246}{5.050} = 6.385 \end{align}\]

8 ANOVA Table

Now we can build our ANOVA table:

	d.f.	CV	SS	MS	F
Treatment (A)	1	5.3177	141.138	141.138	27.948
Type (B)	1	5.3177	480.827	480.827	95.213
Interaction(AxB)	1	5.3177	32.246	32.246	6.385
Within (error)	8		40.404	5.050
Total	11		694.615

So based on that we can see that all three factors cross the \(\alpha = 0.05\) threshold. Doing the operation in R confirms this.

9 Do It In R

co2 <- co2[,.(meanUptake = mean(uptake)), by = .(Plant,Type,Treatment)]
head(co2)

##    Plant   Type  Treatment meanUptake
## 1:   Qn1 Quebec nonchilled   33.22857
## 2:   Qn2 Quebec nonchilled   35.15714
## 3:   Qn3 Quebec nonchilled   37.61429
## 4:   Qc1 Quebec    chilled   29.97143
## 5:   Qc2 Quebec    chilled   32.70000
## 6:   Qc3 Quebec    chilled   32.58571

co2.model = aov(meanUptake ~ Type + Treatment + Type:Treatment, data = co2)
co2.model

## Call:
##    aov(formula = meanUptake ~ Type + Treatment + Type:Treatment, 
##     data = co2)
## 
## Terms:
##                     Type Treatment Type:Treatment Residuals
## Sum of Squares  480.7906  141.1592        32.2471   40.4045
## Deg. of Freedom        1         1              1         8
## 
## Residual standard error: 2.247345
## Estimated effects may be unbalanced

anova(co2.model)

## Analysis of Variance Table
## 
## Response: meanUptake
##                Df Sum Sq Mean Sq F value    Pr(>F)    
## Type            1 480.79  480.79 95.1955  1.02e-05 ***
## Treatment       1 141.16  141.16 27.9492 0.0007402 ***
## Type:Treatment  1  32.25   32.25  6.3849 0.0354301 *  
## Residuals       8  40.40    5.05                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Yeah, that’s much faster.

10 F-Distribution Table \(\alpha\) = 0.05

/	dfN = 1	2	3	4	5	6	7	8	9	10	12	15	20	24	30	40	60	120	∞
dfD=1	161.4476	199.5000	215.7073	224.5832	230.1619	233.9860	236.7684	238.8827	240.5433	241.8817	243.9060	245.9499	248.0131	249.0518	250.0951	251.1432	252.1957	253.2529	254.3144
2	18.5128	19.0000	19.1643	19.2468	19.2964	19.3295	19.3532	19.3710	19.3848	19.3959	19.4125	19.4291	19.4458	19.4541	19.4624	19.4707	19.4791	19.4874	19.4957
3	10.1280	9.5521	9.2766	9.1172	9.0135	8.9406	8.8867	8.8452	8.8123	8.7855	8.7446	8.7029	8.6602	8.6385	8.6166	8.5944	8.5720	8.5494	8.5264
4	7.7086	6.9443	6.5914	6.3882	6.2561	6.1631	6.0942	6.0410	5.9988	5.9644	5.9117	5.8578	5.8025	5.7744	5.7459	5.7170	5.6877	5.6581	5.6281
5	6.6079	5.7861	5.4095	5.1922	5.0503	4.9503	4.8759	4.8183	4.7725	4.7351	4.6777	4.6188	4.5581	4.5272	4.4957	4.4638	4.4314	4.3985	4.3650

6	5.9874	5.1433	4.7571	4.5337	4.3874	4.2839	4.2067	4.1468	4.0990	4.0600	3.9999	3.9381	3.8742	3.8415	3.8082	3.7743	3.7398	3.7047	3.6689
7	5.5914	4.7374	4.3468	4.1203	3.9715	3.8660	3.7870	3.7257	3.6767	3.6365	3.5747	3.5107	3.4445	3.4105	3.3758	3.3404	3.3043	3.2674	3.2298
8	5.3177	4.4590	4.0662	3.8379	3.6875	3.5806	3.5005	3.4381	3.3881	3.3472	3.2839	3.2184	3.1503	3.1152	3.0794	3.0428	3.0053	2.9669	2.9276
9	5.1174	4.2565	3.8625	3.6331	3.4817	3.3738	3.2927	3.2296	3.1789	3.1373	3.0729	3.0061	2.9365	2.9005	2.8637	2.8259	2.7872	2.7475	2.7067
10	4.9646	4.1028	3.7083	3.4780	3.3258	3.2172	3.1355	3.0717	3.0204	2.9782	2.9130	2.8450	2.7740	2.7372	2.6996	2.6609	2.6211	2.5801	2.5379

11	4.8443	3.9823	3.5874	3.3567	3.2039	3.0946	3.0123	2.9480	2.8962	2.8536	2.7876	2.7186	2.6464	2.6090	2.5705	2.5309	2.4901	2.4480	2.4045
12	4.7472	3.8853	3.4903	3.2592	3.1059	2.9961	2.9134	2.8486	2.7964	2.7534	2.6866	2.6169	2.5436	2.5055	2.4663	2.4259	2.3842	2.3410	2.2962
13	4.6672	3.8056	3.4105	3.1791	3.0254	2.9153	2.8321	2.7669	2.7144	2.6710	2.6037	2.5331	2.4589	2.4202	2.3803	2.3392	2.2966	2.2524	2.2064
14	4.6001	3.7389	3.3439	3.1122	2.9582	2.8477	2.7642	2.6987	2.6458	2.6022	2.5342	2.4630	2.3879	2.3487	2.3082	2.2664	2.2229	2.1778	2.1307
15	4.5431	3.6823	3.2874	3.0556	2.9013	2.7905	2.7066	2.6408	2.5876	2.5437	2.4753	2.4034	2.3275	2.2878	2.2468	2.2043	2.1601	2.1141	2.0658

16	4.4940	3.6337	3.2389	3.0069	2.8524	2.7413	2.6572	2.5911	2.5377	2.4935	2.4247	2.3522	2.2756	2.2354	2.1938	2.1507	2.1058	2.0589	2.0096
17	4.4513	3.5915	3.1968	2.9647	2.8100	2.6987	2.6143	2.5480	2.4943	2.4499	2.3807	2.3077	2.2304	2.1898	2.1477	2.1040	2.0584	2.0107	1.9604
18	4.4139	3.5546	3.1599	2.9277	2.7729	2.6613	2.5767	2.5102	2.4563	2.4117	2.3421	2.2686	2.1906	2.1497	2.1071	2.0629	2.0166	1.9681	1.9168
19	4.3807	3.5219	3.1274	2.8951	2.7401	2.6283	2.5435	2.4768	2.4227	2.3779	2.3080	2.2341	2.1555	2.1141	2.0712	2.0264	1.9795	1.9302	1.8780
20	4.3512	3.4928	3.0984	2.8661	2.7109	2.5990	2.5140	2.4471	2.3928	2.3479	2.2776	2.2033	2.1242	2.0825	2.0391	1.9938	1.9464	1.8963	1.8432

21	4.3248	3.4668	3.0725	2.8401	2.6848	2.5727	2.4876	2.4205	2.3660	2.3210	2.2504	2.1757	2.0960	2.0540	2.0102	1.9645	1.9165	1.8657	1.8117
22	4.3009	3.4434	3.0491	2.8167	2.6613	2.5491	2.4638	2.3965	2.3419	2.2967	2.2258	2.1508	2.0707	2.0283	1.9842	1.9380	1.8894	1.8380	1.7831
23	4.2793	3.4221	3.0280	2.7955	2.6400	2.5277	2.4422	2.3748	2.3201	2.2747	2.2036	2.1282	2.0476	2.0050	1.9605	1.9139	1.8648	1.8128	1.7570
24	4.2597	3.4028	3.0088	2.7763	2.6207	2.5082	2.4226	2.3551	2.3002	2.2547	2.1834	2.1077	2.0267	1.9838	1.9390	1.8920	1.8424	1.7896	1.7330
25	4.2417	3.3852	2.9912	2.7587	2.6030	2.4904	2.4047	2.3371	2.2821	2.2365	2.1649	2.0889	2.0075	1.9643	1.9192	1.8718	1.8217	1.7684	1.7110

26	4.2252	3.3690	2.9752	2.7426	2.5868	2.4741	2.3883	2.3205	2.2655	2.2197	2.1479	2.0716	1.9898	1.9464	1.9010	1.8533	1.8027	1.7488	1.6906
27	4.2100	3.3541	2.9604	2.7278	2.5719	2.4591	2.3732	2.3053	2.2501	2.2043	2.1323	2.0558	1.9736	1.9299	1.8842	1.8361	1.7851	1.7306	1.6717
28	4.1960	3.3404	2.9467	2.7141	2.5581	2.4453	2.3593	2.2913	2.2360	2.1900	2.1179	2.0411	1.9586	1.9147	1.8687	1.8203	1.7689	1.7138	1.6541
29	4.1830	3.3277	2.9340	2.7014	2.5454	2.4324	2.3463	2.2783	2.2229	2.1768	2.1045	2.0275	1.9446	1.9005	1.8543	1.8055	1.7537	1.6981	1.6376
30	4.1709	3.3158	2.9223	2.6896	2.5336	2.4205	2.3343	2.2662	2.2107	2.1646	2.0921	2.0148	1.9317	1.8874	1.8409	1.7918	1.7396	1.6835	1.6223

40	4.0847	3.2317	2.8387	2.6060	2.4495	2.3359	2.2490	2.1802	2.1240	2.0772	2.0035	1.9245	1.8389	1.7929	1.7444	1.6928	1.6373	1.5766	1.5089
60	4.0012	3.1504	2.7581	2.5252	2.3683	2.2541	2.1665	2.0970	2.0401	1.9926	1.9174	1.8364	1.7480	1.7001	1.6491	1.5943	1.5343	1.4673	1.3893
120	3.9201	3.0718	2.6802	2.4472	2.2899	2.1750	2.0868	2.0164	1.9588	1.9105	1.8337	1.7505	1.6587	1.6084	1.5543	1.4952	1.4290	1.3519	1.2539
∞	3.8415	2.9957	2.6049	2.3719	2.2141	2.0986	2.0096	1.9384	1.8799	1.8307	1.7522	1.6664	1.5705	1.5173	1.4591	1.3940	1.3180	1.2214	1.0000