Relationship Between t and F tests

Relationship Between t and F tests

In comparing two sample means, paired or unpaired, the F test used in an AOV leads to the same statistical conclusion as the t-tests discussed in Chapter 6. To illustrate the equivalency of an F test and a t-test for unpaired samples, the data of Table 6-1 are again given in Table 7-7.

Table 7-7. Sugar beet root yields for two nitrogen treatments.

Treatment(Ib N/acre)	Replications (tons/acre)					Total (Yi)	Mean (Ȳi)
50	38.8	37.6	37.4	35.8	34.4	188.0	37.6
100	40.6	39.2	39.5	38.6	39.8	198.0	39.6
						Y..=386	Ȳ..=38.60

The AOV of this data is shown in Table 7-8.

Table 7-8. AOV of data in Table 7-7

Source	df	SS		MS		F
Total	9		18.26
Nitrogen	1		10	10			9.71
Experimentall error	8	8.26		1.03

C = 386.0²/10 = 14.899.60
TSS = 38.8² + 37.6² + ... + 39.8² - C
 = 14,917.86 - 14.899.60 = 18.26  

 STT = (188.0² + 198.0²)/5 - C 

  = 14,909.60 - 14,899.60
 = 10.00 with df = 1 

 SSE = TSS - SST = 18.26 - 10.00 = 8.26 with df = 8

 Mean squares are obtained by dividing the sums of squares by their corresponding degrees of freedom. The F test of the equality of the two means is:

F = MST/MSE = 10.00/1.03 = 9.71

which is significant at the 5% level (F_1,8 = 5.32).
            Recall the t-test shown in section 1 of chapter 6 on the same data, has a t-value, 3.13 and t2 = 9.80. Allowing for rounding errors this equals the above calculated F. In fact, for any given α and df can be verified mathematically as follows:

t²_df = [ȳ₁ - ȳ₂ / S_d̄]² 

=2rΣ(ȳ₁ - ȳ_......)²/ rS_d̄ 

₌2rΣ(ȳ₁ - ȳ_......)²/ r(2S²/r)

=MST/MSE 

=F_1,df 

It can be shown by comparing F_1,df values in Appendix Table A-7 with the T-values in Appendix Table A-6. For instance, the 5% critical t-value with 8 df is 2.306 and thus  t²_df =5.32 = F₁₈ Therefore the two tests are equivalent, since identical conclusion will be drawn.

SUMMARY

1. The analysis of variance is the most powerful technique to analyze well-designed experiments. The completely randomized design is the simplest experimental design. The data obtained from such designs can be analyzed by the one-way analysis of variance method if the assumptions are met.
2. Assumptions underlying the analysis of variance: 
 
1. Experimental errors (or observations) are normally distributed.
2. Experimental errors (or observations) are independently distributed.
3. Experimental errors (or observations) from different treatments have the same variance.
4. Additivity between treatment and environmental effects. 
3.  AOV of CRD without subsamples: for k treatments and r replications per treatment,
                       

Source of variation	df	Sum of square		Mean Square		F
Total	Kr-1		ΣΣY2ij - C
Between treatments	k-1		ΣY2i / r-C	SST/(k-1)			MST/MSE
Experimental  error	K(r-1)	ΣΣ(Y2i / r)		SSE/(r-1)

If not all treatments have equal replications, and r₁ is the replication number of treatments, i, then  SST = ΣY²_i / r_i -C and all other calculations are the same

4. AOV of CRD with subsamples: For k treatments, r replications per treatment and n subsamples per experimental unit.

   Source of variation	df	Sum of square		Mean Square		F
   Total	Krn-1		ΣΣY2ijk - C
   Between exp. units	Kr-1		ΣY2ij / n-C
   Between treatments	k-1		ΣY2i.. / rn-C	SST/(k-1)			MST/MSE
   Experimental  error	K(r-1)	ΣΣY2ij / n - ΣY2i.. / nr		SSE/k(r-1)		MSE/MSS
   Sampling Error	Kr(n-1)	ΣΣΣ2ijk- ΣΣ2ij/ n		SSS/kr(n-1)

    
   
   When subsamples are taken, experimental error consists of two sources of variation, variation among subsamples, σ²_s , and variation among units, σ²_e , treated alike, i.e., MSE is an estimate of  σ²_s + nσ²_e The estimate of is MSS. The σ²_e  is (MSE - MSS)/n which measures the variation among means of similarly treated units over and above that due to sub samples within units. A test of the hypothesis available by comparing F = MSE/MSS with critical F-values. e


5. F_1,df  = t²_df where df is the degrees of freedom of the variance estimate.