PRINCIPLES OF EXPERIMENTAL DESIGN
Introduction
In designing an experiment, it is essential to state the objectives of the experiment as to answer the questions, stated hypothesis to be tested and the effect to be estimated.Experimental design is how treatments under investigation are arranged such that their effect are revealed and are accurately measured. All designs are characterized by experimental units classified by treatments, but in some cases they are further classified into blocks, rows, columns main plots and so on. An experimental design can be complex or simple.
Objective
To evaluate the information in the principles of the experimental design.
TOPIC 1: EXPERIMENT
Important Content
Experiment is an investigation to obtain
a)new information
b)proving the result of an earlier experiment
c) itis conducted to answer question(s)
TOPIC 2: TREATMENT
Important Content
Procedure whose effect of a material to be tested and compared with other treatments
Example 1:types of fertilizer: NPK Blue and NPK yellow
Example 2:fertilizer rates: 10, 20 and 30 kg N ha-1
TOPIC3: EXPERIMENTAL UNIT
Important Content
This is the unit of material that receives a treatment or where treatment is given. The unit may be a single plant, a single animal, a leaf, ten plants in a square meter plot or so on. For a field experiments, a decision on the size and shape of the experimental unit has to be made. In situations where non-uniformity in experimental plot is anticipated, the plots should be reasonably long and narrow. Effort should be made to control the influence of each adjacent unit on the other. This can be achieved by randomizing treatments and also by making use of guard rows.
Example:a plant
an animal
area of land
a square meter plot
Example 1: Effect of different type of fertilizers on sunflower
Fertilizer NPK BlueFertilizer NPK Yellow
Experimental unit
(a plant)
Example 2: Effect of fertilizer rates on a plant
10 kg N ha-120 kg N ha-1
Experimental unit
(an area)
TOPIC 4: SAMPLE
Important Content
Part of experimental unit where the effect of treatment is measured
10 kg N ha-120 kg N ha-1
Sample (only these part is measured)
TOPIC 5: REPLICATION
ImportantContent
Repetitionor appearance of a treatment more than once in an experiment is referred to as replication. Replication is the sole means of measuring the validity of a conclusion drawn from the experiment, the number of replications should be chosen such that the required precision of the treatment estimate is produced.Several factors affect the number of replications for an experiment; perhaps the most important of all is the degree of precision required. When a treatment effect is small and requires high precision to be detected or measured, the greater the number of replicates the better.
10 kg Nha-120 kg N ha-1
Replication 1
10 kg N ha-120 kg N ha-1
Replication2
Purpose of replication:
• to calculate the mean of the treatment
• to improve the accuracy of the experiment
• to measure the experimental error
TOPIC 6: RANDOMIZATION
Important Content
Arrangement of treatments of experimental unit so as that each experimental unit has the same chance to be selected to receive a treatment
Example : Effect of 4 types of fertilizer with 2 replications.
A / BC / D
REPLICATION 1
D / CA / B
REPLICATION 2
TOPIC 7: VARIABLES
Important Content
Characteristics of the experimental unit that can be measured:
• Yield
• Height of a plant
• Soil pH
• Number of insects
2 types of variables
• Quantitative
• Qualitative
TOPIC 8: CONTROL
Important Content
A standard treatment that is used as a baseline or basis of comparison for the other treatments. The control treatment does not receive the treatment or the experimental manipulation that the experimental treatments receive.
TOPIC 9: RESPONSES
Important Content
Outcomes that are being observed after applying a treatment to an experimental unit
Example: Treatment :- application of 3 types of nitrogen fertilizer
Response :- nitrogen content or biomass of corn plants
TOPIC 10: EXPERIMENTAL ERROR
Important Content
The random variation present in all experimental results. Errors can beminimized by having a large sample size as well as by replications and blocking.
TOPIC 11: TYPES OF EXPERIMENT
Important Content
- Manipulative experiment – one or more conditions are varied while all other conditions are maintained, perform under controlled conditions such as in a laboratory
- Field experiment – similar to manipulative experiment, but it is carried out in an open area where environmental factors and extraneous variables are present.
TOPIC 12: SELECTION OF TEST SITE
Important Content
Selection of sites where the trial is to be conducted
Selection procedures:
- Clearly specify the desired test environment and identify the sources of variability e.g. soil, climate, topography, water regime.
- Select a large area that is homogenous and satisfies those selected features mentioned above.
- Choose an area/field that is large enough to accommodate the experiment.
TOPIC 13: UNIFORMITY OF EXPERIMENTAL SITE
Important Content
- Slope - Fertility gradients are more pronounced in sloping areas. Ideally, experiments should be conducted in areas with no slopes (level). If this not avoidable, proper blocking is needed.
- Areas used for experiments in previous cropping - When the area to be used for a future experiment has been used in a previous experiment, study the nature of the previous study to determine if it will have any direct or serious effect on the outcome of the new experiment.
- Presence of large trees, poles, and structures - Areas with surrounding permanent structures should be avoided, not only because of the direct effect of shading but also the nature of the soil near the structure.
TOPIC 14: PROCEDURES IN PLANNING AN EXPERIMENT
Important Content
- Statement of the objectives of the experiment
- Identification of the resources available for the experiment
- Assessment of the location and the conditions under which the experiment to be conducted
- Identification of the population of subjects that are to be tested
- Consideration of the amount of variability that is likely to arise within samples
- Identification of the type of observation/measurements that are to be made
- Identification of the most appropriate technique for analyzing data
- Identification of treatment groups and assignment of treatments
TOPIC 15: TYPES OF MEASUREMENT/DATA
Important Content
A response or dependent variable that really provides information about the problem under study
Primary observations = grain yield
Explanatory observations = number of tillers, panicle number, spikelet number
Covariate observations = percent infestation (if the plants were infested by disease)
TOPIC 16: HYPOTHESIS TESTING
Important Content
It is a statistical test used to the objective
Two types of hypotheses:
Null hypothesis (H0)
Alternative hypothesis (HA)
Significance testing is achieved based on the critical level of probability which is commonly set to 5% or α=0.05, written as (P≤ 0.05).
H0 = there is no difference between the sample means µ1= µ2 = µn
HA = there is a difference between the sample means µ1 ≠ µ2 ≠ µn
If the value of P is smaller than (or equal to) the critical value (α=0.05), H0 is rejected while HA is accepted.
If the value of P is larger than the critical value, H0 is accepted while HA is rejected.
TOPIC 17: METHODS OF ERROR CONTROL IN EXPERIMENT
Important Content
- Blocking
- Proper plot technique
- Covariance analysis
TOPIC 18: PLOT SIZE AND SHAPE
Important Content
An experiment conducted on soils of high variability require a small plot size with increased number of replications will minimize or reduce experimental error because the distance between any farthest points in each block will be shorter than when using large plots. As a result, the variability within each block is minimized.
If the plot size cannot be reduced and it is suspected that the soil is highly variable with unknown direction, the use of square-shaped blocks is recommended. The distance between any two farthest points in a square block is shorter that those in along and narrow block.
TOPIC 19: UNIFORMITY OF EXPERIMENT PLOT
Important Content
In a plot, each block must be of the same size and shape with equal numbers of experimental units arranged randomly according to the specified design. Except for split plot design, the size of the main plot is bigger than the sub-plot size.
EXPERIMENTAL DESIGNS
Introduction
Arrangement of experimental unit that contains treatments and replications into various designs to estimate and control experimental error so as to interpret results accurately.The major difference among experimental designs is the way in which experimental units are classified or grouped. An experimental design can be simple or complex. It is, however, advisable to choose a less complicated design that best provides the desired precision.
Objective
To estimate and control experimental error for accurate interpretation,
TOPIC 1:COMPLETE RANDOMIZED DESIGN
Important Content
It is used when an area or location or experimental materials are homogeneous. For completely randomized design (CRD), each experimental unit has the same chance of receiving a treatment in completely randomized manner.
Example: Testing 4 varieties (V1, V2, V3 and V4) in a homogeneous field.
V1 / V2V3 / V4
The soil is homogeneous so the varieties can be located at any of the compartment without any effect of the soil.
All the 4 compartments have the same soil fertility.
.....effect of block is neglected or is not considered
....easy placement as the treatment can be placed in any of the compartments
....easy to arrange experimental unit due to lack of block effects
Disadvantage: difficult to obtain homogeneity in the field.
Example: Testing of yield of 4 crop varietieswith 4 replications.
Varieties: V1, V2, V3, V4 (control)
ReplicationsR1, R2, R3, R4
V1 R3 / V2 R2 / V1 R4 / V4 R1V2 R4 / V1 R2 / V3 R1 / V4 R4
V2 R1 / V2 R3 / V4 R2 / V3 R4
V3 R2 / V1 R1 / V4 R3 / V3 R3
All the varieties with 4 replications can be placedat any of the compartments
Each compartment the soil fertility is the same.
TOPIC 2: RANDOMIZED COMPLETE BLOCK DESIGN
Important Content
In this design treatments are assigned at random to a group of experimental units called the block. A block consists of uniform experimental units. The main aim of this design is to keep the variability among experimental units within a block as small as possible and to maximize differences among the blocks.
....it is used for an area or location or materials that are heterogeneous
....group of treatments is placed randomly in a block or replication
....block or replication is created to reduce the heterogeneity of the experimental unit
....each block containing homogenous experimental unit
....treatments are arranged in each block or replication
....effect of block is considered in the calculation of ANOVA
Method of blocking in a field
a) One directional gradient
Arrange the block at right angles with the gradient
High Fertility Low Fertility
b)Two ways gradients: 1 strong, 1 less in strength
Moderate
Arrangeblock perpendicular to the gradient
High Fertility Low Fertility
c)Two ways gradients same strength
Square blocking as much as possible
Example : Testing 4 crop varieties with 4 replications.
Varieties:V1, V2, V3, V4 (control)
Replication:R1, R2, R3, R4
R1 / R2 / R3 / R4V1 / V2 / V3 / V4
V4 / V1 / V1 / V2
V3 / V4 / V4 / V3
V2 / V3 / V2 / V1
Fertility Gradient
TOPIC 3: LATIN SQUARE DESIGN
Important Content
Latin square design handles two known sources of variation among experimental units simultaneously. It treats the sources as two independent blocking criteria: row-blocking and column-blocking. This is achieved by making sure that every treatment occurs only once in each row-block and once in each column-block. This helps to remove variability from the experimental error associated with both these effects.
•Treatments are arranged in row and column
•Error is being reduced due to two ways heterogeneity (row and column)
•More efficient than RCBD when there is two ways heterogeneity
•Number of replication should be equal to number of treatment
•Usually such arrangement is suitable for 4 to 8 treatments
STEPS IN ARRANGING TREATMENTS WITH RANDOMIZATION IN A LATIN SQURE DESIGN
Example: Effect of 6 different fertilizer N treatments (A, B, C, D, E, and F) on the yield of corn.
1.Arrange each treatment so that it occurs once in a row and once in a column only.
Row / ColumnB / D / E / F / A / C
C / E / A / D / F / B
A / F / C / B / E / D
D / A / F / C / B / E
F / B / D / E / C / A
E / C / B / A / D / F
- Use random number table, assign numbering for row and column randomly.
Row / Column
4 / 2 / 5 / 1 / 3 / 6
1 / B / D / E / F / A / C
3 / C / E / A / D / F / B
5 / A / F / C / B / E / D
4 / D / A / F / C / B / E
2 / F / B / D / E / C / A
6 / E / C / B / A / D / F
- Arrange the treatments in the field based on the arrangement in the above table 2.
Row / Column
1 / 2 / 3 / 4 / 5 / 6
1 / F / D / A / B / E / C
3 / E / B / C / F / D / A
5 / D / E / F / C / A / B
4 / C / A / B / D / F / E
2 / B / F / E / A / C / D
6 / A / C / D / E / B / F
ANALYSIS OF VARIANCE
Introduction
Analysis of variance (ANOVA) is to determine the ratio of between samples to the variance of within samples that is the F distribution. The value of F is used to reject or accept the null hypothesis. It is used to analyze the variances of treatments or events for significant differences between treatment variances, particularly in situations where more than two treatments are involved. ANOVA can only be used to ascertain if the treatment differences are significant or not.
Objective
To accept or reject the null hypothesis where more than two treatments are involved.
TOPIC 1: F DISTRIBUTION
Important Content
F value is used to test the significant difference between more than two treatment means
F =s2, calculated from sample mean
s2, calculate from variance between individual sample
=sa2 (variance between samples)
sd2 (variance within samples)
df (numerator) = n -1, where n = number of samples
df (denominator) = n(r – 1), where r = size of samples
TOPIC 2: ANOVA FOR ONE FACTOR EXPERIMENT
Important Content
-Using the same data, F can be calculated using Table of ANOVA:
- Table of ANOVA
Source dfSum of SquaresMean Square F
of Variation(SS) (MS)
Treatment n-1 SS treatment SStreatment/n-1 SStreatment/MSerror
Error n(r-1) SSerror SSerror/n(r-1)
Total rn-1 SSTotal
TOPIC 3: ANOVA FOR VARIOUS DESIGNS
Important Content
- Complete Randomized Design
Example: Testing yield of 4 varieties with 4 replications.
Varieties: V1, V2, V3, V4 (Control)
Replications:R1, R2, R3, R4
V1R3 (50) / V2R2 (69) / V1R4(54) / V4R1(51)V2 R4 (57) / V1R2 (67) / V3R1 (65) / V4R4 (62)
V2R1 (57) / V2R3 (53) / V4R2 (52) / V3R4 (74)
V3R2 (54) / V1R1 (57) / V4R3 (47) / V3R3 (59)
Calculation:
Arrange the data according to treatment and replication
Variety / Replication / Total / Mean1 / 2 / 3 / 4
V1 / 57 / 67 / 50 / 54 / 228 / 57
V2 / 57 / 69 / 53 / 57 / 236 / 59
V3 / 65 / 54 / 59 / 74 / 252 / 63
V4 / 51 / 52 / 47 / 62 / 212 / 53
Total / 928
ANOVA
HO: no significant difference in yield between the varieties.
Table of ANOVA
Source of / df / SS / MS / F / FTableVariation / (p=0.05)
Variety / 3 / 208 / 69.3 / 1.29 / 3.49
Error / 12 / 646 / 53.8
Total / 15 / 854
Calculation
1.Degree of Freedom, df
df (total)= vr – 1 = 4(4) – 1 = 15
df (variety)= v – 1 = 4 – 1 = 3
df (error)= df(total) – df(variety) = 15 – 3 = 12
or
df (error) = v(r – 1) = 4(4 – 1) = 12
2.Correction factor, CF
CF = Y..2 / rv = (928)2/(4×4) = 53824
3.Sum Square Total (SST)
SST= ƩYij2 – CF
= (572 + 672 + ... + 472 + 622) – 53824
= 54678 – 53824 = 854
4.Sum Square Variety (SSV)
SSV = (ƩY.j2/ r) – CF
= (2282 + 2362 + 2522 + 2122)/ 4 – 53824
= 54032 – 53824 = 208
5. Sum Square Error (SSE)
SSE= SST – SSV
= 854 – 208 = 646
6. Mean Square Variety (MSV)
MSV= SSV/dfv = 208/3 = 69.3
7.Mean Square Error (MSE)
MSE= SSE/dfe = 646/ 12 = 53.8
8.F value Variety
F= MSV/MSE = 69.3/53.8 = 1.29
9.F Table
dfv = 3, dfe = 12
At P = 0.05, F = 3.49
10. Conclusion
1.29 < 3.49 →accept HO, no significant difference of yield between the varieties.
- Randomized Complete Block Design
Arrange data according to treatments and replications.
Variety / R1 / R2 / R3 / R4 / Total / MeanV1 / 57 / 67 / 50 / 54 / 228 / 57
V2 / 57 / 69 / 53 / 57 / 236 / 59
V3 / 65 / 54 / 59 / 74 / 252 / 63
V4 / 51 / 52 / 47 / 62 / 212 / 53
Total / 230 / 242 / 209 / 247 / 928
Mean / 57.50 / 60.50 / 52.25 / 61.75
Calculation:
HO: no significant difference of yield between varieties.
Table of ANOVA
Source of / df / SS / MS / F / FTableVariation / (p=0.05)
Block (Rep) / 3 / 214.5 / 71.5 / 1.49 / 3.86
Variety / 3 / 208 / 69.3 / 1.45 / 3.86
Error / 9 / 431.5 / 47.94
Total / 15 / 854
Calculation
1.Degree of Freedom (df)
df (total)= vr – 1 = 4(4) – 1 = 15
df (block)= r – 1 = 4 – 1 = 3
df (variety)= v – 1 = 4 – 1 = 3
df (error)= df (total) – df(block) – df(variety)
= 15 – 3 – 3 = 9 Or
df (error)= (v-1)(r-1) = (4-1)(4-1) = 9
2. Correction Factor, CF
CF = Y..2/rv = (928)2/ (4×4) = 53824
3. Sum Square Total (SST)
SST= ƩYij2 – CF
= (572 + 672 + ... + 472 + 622) – 53824
= 54678 – 53824 = 854
4. Sum Square Block (SSB)
SSB = (Ʃy.j2/v) – CF
= (2302 + 2422 + 2092 + 2472)/ 4 – 53824
= 54038.5 – 53824 = 214.5
5. Sum Square Variety (SSV)
SSV= (ƩYi.2/r) – CF
= (2282 + 2362 + 2522 + 2122)/4 – 53824
= 54032 – 53824 = 208
6.Sum Square Error (SSE)
SSE = SST – SSB – SSV
= 854 – 214.5 – 208 = 431.5
7.Mean Square Blok (MSB)
MSB = SSB/dfb = 214.5/3 = 71.5
8.Mean Square Variety (MSV)
MSV = SSV/dfv = 208/3 = 69.3
9.Mean Square Error (MSE)
MSE= SSE/dfe = 431.5/9 = 47.94
10. F value
F value (block) = MSB/MSE = 71.5/47.94 = 1.49
F (variety) = MSV/MSE = 69.3/47.94 = 1.45
11.F Table
Block: dfb = 3, dfe = 9, atp = 0.05, F = 3.86
Variety: dfv = 3, dfe = 9, atp = 0.05, F = 8.91
12. Conclusion
Variety: 1.45 < 3.86 → acceptHO, there is no significant different between varieties on yield.
Block: 1.49 < 3.86 → acceptHO, there is no significant effect of block on the yield.
- Latin Square Design
1.Arrange the data according to treatment as the arrangement in the experiment whereby the treatment occur once in the column and once in the row.
ROW / COLUMN1 / 2 / 3 / 4 / 5 / 6 / Total
1 / A(32.1) / B(33.1) / C(32.4) / D(29.1) / E(31.1) / F(28.2) / 186.0
2 / F(24.8) / A(30.6) / B(29.5) / C(29.4) / D(33.0) / E(31.0) / 178.3
3 / E(28.8) / F(21.7) / A(31.9) / B(30.1) / C(30.8) / D(30.6) / 173.9
4 / D(31.4) / E(31.9) / F(26.7) / A(30.4) / B(28.8) / C(33.1) / 182.3
5 / C(33.5) / D(32.3) / E(30.3) / F(25.8) / A(30.3) / B(30.7) / 182.9
6 / B(30.7) / C(29.7) / D(27.4) / E(29.1) / F(21.4) / A(30.8) / 169.10
Total / 181.3 / 179.3 / 178.2 / 173.9 / 175.4 / 184.40
]
2. Calculate ANOVA.
Source of Variation / df / SS / MS / F / F(0.05)Row / 5 / 33.20 / 6.640 / 2.63 / 2.71
Column / 5 / 12.29 / 2.458 / 0.98 / 2.71
Treatment / 5 / 186.78 / 37.356 / 14.81 / 2.71
Error / 20 / 50.43 / 2.521
Total / 35 / 282.70
3. DegreeFreedom (df)
dfT (total) = rc -1 = 6(6)-1 = 35
dfR (row) = r – 1 = 6-1 = 5
dfC (column) = c -1 = 6-1 = 5
dfV (treatment) = b -1 = 6-1 = 5
dfE (error) = DfT – DfR – DfC – DfB = 20
or= (r-1)(c-1) – (v-1) = (5)(5) – 5 = 20
4.Correction Factor, CF
CF = Y...2 / rc
= (1072.5)2 / 6(6)
= 31951.56
5. Sum of Square (SS) and Mean Square (MS)
Total SST =ƩY2… – CF
= 32.12 + 33.12 + .... + 21.42 + 30.82 – 31951.56
=282.70
RowSSR=Ʃyi..2/c – CF
=(186.02 +.... + 169.12)/6 – 31903.91
=33..20
MSR=SSR/DfR= 33.20/5
= 6.64
Column SSC=Ʃy.j.2/r – CF
=(181.32 + ... + 184.42)/6 – 31951.56
=12.29
MSC=SSC/DfC= 12.29/5
=2.458
Treatment (V)
SSV=Ʃy..K2/rep – CF
=(186.12 + ... + 148.62)/6 – 31951.56
=186.78
MSB=SSV/DfV=186.78/5
=37.356
Error SSE = SST – SSR – SSC – SSV
=282.70 – 33.20 – 12.29 – 186.78
= 50.43
MSE=SSE/DFE=50.43/20
=2.521
6. F value
F (row)=MSR/MSE = 6.640/2.521 = 2.63
F (Column)=MSC/MSE = 2.458/2.521 = 0.98
F (Treatment)=MSV/MSE = 37.356/2.521 =14.81
7.Table F value
Rows:dfR = 5, dfE = 20, p = 0.05 F = 2.71
Column:dfC = 5, dfE = 20, p = 0.05 F = 2.71
Treatment:dfV = 5, dfE = 20, p = 0.05 F = 2.71
Conclusion
Treatment:F (14.81) > F table (2.71) Reject HO, there is at least one significant difference between the treatments.
Row:F value (2.63) F table (2.71) acceptHO, there is no significant difference between the rows.
Column: F value (0.98)F table (2.71) acceptHO, there is no significant different between columns.
COMPARISON OF TREATMENT MEANS
Introduction
Comparison of means is conducted when the null hypothesis (HO) is being rejected during the process of ANOVA. When HO is rejected, there is at least one significant difference between the treatment means. There are various methods to compare for significant difference between the treatments means. The means of more than two means are often compared for significant difference using Least Significant Difference (LSD) test, Duncan’s New Multiple Range (DMRT) test, Tukey’s test, Scheffe’s test, Student –Newman-Keul’s test (SNK), Dunnett’s test and Contrast. However, more often than not, such tests are misused. One of the main reasons for this is the lack of clear understanding of what pair and group comparisons as well as what the structure of treatments under investigation are. There are two types of pair comparison namely planned and unplanned pair.
Objective
To compare between the treatment means after rejecting the HO from ANOVA.
TOPIC 1: Least Significant Difference (LSD)
ImportantContent
It is a t-test and usually suitable to compare between two means.
Example:
Source of / df / SS / MS / F / FtableVariation / (p=0.05)
Block (Rep) / 3 / 576 / 192 / 24.7 / 3.86
Variety / 3 / 208 / 69.3 / 8.9 / 3.86
Error / 9 / 70 / 7.78
Total / 15 / 854
Arrange the means from low to high or from high to low
Variety / V4 / V1 / V2 / V3Mean / 53 / 57 / 59 / 63
Calculation
T = (d - µd) / sd
D = Y1 – Y2
Assume the mean are from the same population, so µd = 0
t = d/sd
t = LSD/ sd
LSD = t. sd
sd = √2MSE/r
sd = √2(7.78) /4 = 1.972
obtain t value from table df = dfe = 9, p = 0.05
t = 2.262
LSD = 2.262 × 1.972 = 4.46 t ha-1
Compare the difference of two means and compare with the LSD value,
Higher than LSD value→ significant different
Lower than LSD value→no significant difference
V3– V2 = 63- 59 = 4, 4.46 → no significant difference
V3 – V1 = 63 – 57 = 6, 4.46 → significant different
V3 – V4 = 63 – 53 =10, >4.46→ significant different
V2 – V1 = 59 – 57 = 2, 4.46 → no significant difference
V2 – V4 = 59 – 53 = 6, 4.46 → significant different
V1– V4 = 57 – 53 = 4, 4.46 → no significant difference
or can be present as the following:
Variety / Yield (tha-1)V3 / 63 a
V2 / 59 ab
V1 / 57 bc
V4 / 53 c
LSD0.05 / 4.46
TOPIC 2: DUNCAN’S MULTIPLE RANGE TEST
Important Content
To compare between treatments means for multiple comparison.
Calculation
1.Calculate LSD value
LSD0.05 = t √2MSE/r = 2.262√2(7.78)/4 = 4.46
2.Calculate D value
D = R(LSD)
R from table, that is up to 4 levels of comparison