©Richard Lowry, 1999-
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raw data | B | ||
0 units | 1 unit | ||
A | 0 units |
20.4 17.4 20.0 18.4 24.5 21.0 19.7 22.3 17.3 23.3 |
20.5 26.3 26.6 19.8 25.4 28.2 22.6 23.7 22.5 22.6 |
1 unit |
22.4 19.1 22.4 25.4 26.2 25.1 28.8 21.8 26.3 25.2 |
34.1 21.9 32.6 28.5 29.0 25.8 29.0 27.1 25.7 24.4 | |
summary data | B | ||||
0 units | 1 unit | rows | |||
A | 0 units |
N_{g1}=10 ∑X_{g1}=204.3 ∑X^{2}_{g1}=4226.3 |
N_{g2}=10 ∑X_{g2}=238.2 ∑X^{2}_{g2}=5741.4 |
N_{r1}=20 ∑X_{r1}=442.5 | |
1 unit |
N_{g3}=10 ∑X_{g3}=242.7 ∑X^{2}_{g3}=5961.34 |
N_{g4}=10 ∑X_{g4}=278.1 ∑X^{2}_{g4}=7855.3 |
N_{r2}=20 ∑X_{r2}=520.8 | ||
columns |
N_{c1}=20 ∑X_{c1}=447.0 |
N_{c2}=20 ∑X_{c2}=516.3 |
N_{T}=40 ∑X_{T}=963.3 ∑X^{2}_{T}=23784.4 |
means | B | ||||
0 units | 1 unit | rows | |||
A | 0 units | M_{g1}=20.43 |
M_{g2}=23.82 |
M_{r1}=22.13 | |
1 unit | M_{g3}=24.27 | M_{g4}=27.81 | M_{r2}=26.04 | ||
columns | M_{c1}=22.35 | M_{c2}=25.82 | M_{T}=24.08 |
SS = ∑X^{2}_{i} — | (∑X_{i})^{2} N |
preliminary SS values | B | |||
0 units | 1 unit | |||
A | 0 units | SS_{g1}=52.44 |
SS_{g2}=67.48 | |
1 unit | SS_{g3}=71.02 | SS_{g4}=121.37 | ||
SS_{T}=585.70 |
SS_{wg} | = SS_{g1} + SS_{g2} + SS_{g3} + SS_{g4} | |
= 52.44 + 67.48 + 71.02 + 121.37 | ||
= 312.31 |
SS_{bg} | = SS_{T} — SS_{wg} | |
= 585.70 — 312.31 | ||
= 273.39 |
SS_{bg} | = | (∑X_{g1})^{2} N_{g1} | + | (∑X_{g2})^{2} N_{g2} | + | (∑X_{g3})^{2} N_{g3} | + | (∑X_{g4})^{2} N_{g4} | — | (∑X_{T})^{2} N_{T} | |
= | (204.3)^{2} 10 | + | (238.2)^{2} 10 | + | (242.7)^{2} 10 | + | (278.1)^{2} 10 | — | (963.3)^{2} 40 | ||
= | 273.39 |
row 1 | row 2 | |
observed row mean | 22.13 | 26.04 |
expected row mean | 24.08 | 24.08 |
deviate | —1.95 | +1.96 |
squared deviate | 3.80 | 3.84 |
squared deviate weighted by number in row (20) | 76.0 | 76.8 |
col 1 | col 2 | |
observed column mean | 22.35 | 25.82 |
expected column mean | 24.08 | 24.08 |
deviate | —1.73 | +1.74 |
squared deviate | 2.99 | 3.03 |
squared deviate weighted by number in column (20) | 59.8 | 60.6 |
SS_{bg} | = | (∑X_{g1})^{2} N_{g1} | + | (∑X_{g2})^{2} N_{g2} | + | (∑X_{g3})^{2} N_{g3} | + | (∑X_{g4})^{2} N_{g4} | — | (∑X_{T})^{2} N_{T} |
SS_{rows} | = | (∑X_{r1})^{2} N_{r1} | + | (∑X_{r2})^{2} N_{r2} | — | (∑X_{T})^{2} N_{T} |
Click here if you would like a printable summary of the data on which these calculations are based. | ||
= | (442.5)^{2} 20 | + | (520.8)^{2} 20 | — | (963.3)^{2} 40 | ||||
= | 153.27 |
SS_{cols} | = | (∑X_{c1})^{2} N_{c1} | + | (∑X_{c2})^{2} N_{c2} | — | (∑X_{T})^{2} N_{T} | |
= | (447.0)^{2} 20 | + | (516.3)^{2} 20 | — | (963.3)^{2} 40 | ||
= | 120.06 |
SS_{rxc} | = SS_{bg} — SS_{rows} — SS_{cols} | |
= 273.39 — 153.27 — 120.06 | ||
= 0.06 |
M_{g*} = | the mean of any particular one of the the individual groups of measures | |
M_{r*} = | the mean of the row to which that group belongs | |
M_{c*} = | the mean of the column to which that group belongs |
[null]M_{g*} = M_{r*} + M_{c*} — M_{T} | Click here for a brief account of the logic of this formula. |
[null]M_{g1} | = M_{r1} + M_{c1} — M_{T} | |
= 22.13 + 22.35 — 24.08 | ||
= 20.40 |
[null]M_{g2} | = M_{r1} + M_{c2} — M_{T} | |
= 22.13 + 25.82 — 24.08 | ||
= 23.87 |
means | B | ||||
0 units | 1 unit | rows | |||
A | 0 units | 20.43 20.40 | 23.82 23.87 |
M_{r1}=22.13 | |
1 unit | 24.27 24.31 | 27.81 27.78 | M_{r2}=26.04 | ||
columns | M_{c1}=22.35 | M_{c2}=25.82 | M_{T}=24.08 |
g1 | g2 | g3 | g4 | |
observed group mean | 20.43 | 23.82 | 24.27 | 27.81 |
expected group mean | 20.40 | 23.87 | 24.31 | 27.78 |
deviate | +0.03 | —0.05 | —0.04 | +0.03 |
squared deviate | 0.0009 | 0.0025 | 0.0016 | 0.0009 |
squared deviate weighted by number in group (10) | 0.009 | 0.025 | 0.016 | 0.009 |
Total: | SS_{T} = 585.70 | |
within groups: | SS_{wg} = 312.31 | |
between groups: | SS_{bg} = 273.39 | |
rows: | SS_{rows} = 153.27 | |
columns: | SS_{cols} = 120.06 | |
interaction: | SS_{rxc} = 0.06 |
df_{rxc} = (r—1)(c—1) | r = number of rows c = number of columns |
in general | for the present example | ||
Total | df_{T} = N_{T}—1 | 40—1=39 | Note that_{T} df_{T}=df_{wg}+df_{bg} |
within- groups (error) | df_{wg} = N_{T}—rc | 40—(2)(2)=36 | |
between- groups | df_{bg} = rc—1 | (2)(2)—1=3 | |
rows | df_{rows} = r—1 | 2—1=1 | Note that_{T} df_{bg}=df_{rows}+df_{cols}+df_{rxc} |
columns | df_{cols} = c—1 | 2—1=1 | |
interaction | df_{rxc} = (r—1)(c—1) | (2—1)(2—1)=1 |
MS_{rows} | = | SS_{rows} df_{rows} | MS_{cols} | = | SS_{cols} df_{cols} | MS_{rxc} | = | SS_{rxc} df_{rxc} | |||
= | 153.27 1 | = | 120.06 1 | = | 0.06 1 | ||||||
= | 153.27 | = | 120.06 | = | 0.06 | ||||||
MS_{error} | = | SS_{wg} df_{wg} | ||
= | 312.31 36 | = 8.68 |
F_{rows} | = | MS_{rows} MS_{error} | F_{cols} | = | MS_{cols} MS_{error} | F_{rxc} | = | MS_{rxc} MS_{error} | |||
= | 153.27 8.68 | = | 120.06 8.68 | = | 0.06 8.68 | ||||||
= | 17.67 | = | 13.84 | = | 0.01 | ||||||
with df=1,36 | with df=1,36 | with df=1,36 |
df denomi- nator | df numerator | |||
1 | 2 | 3 | ||
36 | 4.11 7.40 | 3.26 5.25 | 2.87 4.38 | |
The fundamental meaning of the significant row and column effects is that the difference between the two row means
col 1 [B=0] | col 2 [B=1] | row means | |
row 1 [A=0] | 22.13 | ||
row 2 [A=1] | 26.04 | ||
column means | 22.35 | 25.82 |
But for the present example it is all plain and simple. Each of the two drugs appears to increase arousal, and there is no indication that they interact with each other. When presented in combination, their effects are merely additive.
Source | SS | df | MS | F | P | |
between groups | 273.39 | 1 | ||||
rows | 153.27 | 1 | 153.27 | 17.67 | <.01 | |
columns | 120.06 | 1 | 120.06 | 13.84 | <.01 | |
interaction | 0.06 | 1 | 0.06 | 0.01 | ns | |
within groups (error) | 312.31 | 36 | 8.68 | |||
TOTAL | 585.70 | 39 | ||||
"ns" = "non-significant" |
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