Experiments Using Various Designs

Instructions

Review the description of the experimental design then identify the response variable or variables, treatment factors, sources of blocking, covariables, fixed and random effects, and experimental, sampling, and measurement units. Draw a schematic or picture that describes the experimental layout. Write a mathematical model describing the response variable or variables as a function of the design elements. Describe the response variable or variables as to a reasonable distribution for modeling the response (i.e., normal, binomial, Poisson, etc.) and transformations that might be required. Classify the design as nearly as possible to one of the "named" designs. Identify the treatment structure used. Write an ANOVA source table, if appropriate, giving sources of variation, degrees of freedom, and expected mean squares corresponding with your model. Write out, in terms of the experiment, the hypotheses that the investigator would be interested in, then translate each into a mathematical hypothesis. Which hypotheses would best be investigated via linear contrasts? Orthogonal polynomials? Multiple comparisons? Full versus reduced models?

Experiments

  1. An agronomist wanted to determine the effect of different sources of nitrogen on the yield of barley used as a forage crop. There were five sources to be compared: (1) (NH4)2SO4, (2) NH4NO3, (3) CO(NH2)2, (4) Ca(NO3)2, and (5) NaNO3. He also decided to use a control treatment with no nitrogen. Because he wanted to apply the results over a fairly wide range of conditions, he decided to conduct the trial on four types of soil. He located six plots on each of the four soil types. He then randomly assigned the treatments to the plots within each soil type. The yields were measured in kg/plot. (Source: Petersen 1994: 55)

  2. A horticulturalist wanted to study the effect of row spacing and phosphate fertilizer on the yield of bush beans. She used three row spacings: (S1) 45 cm, (S2) 90 cm, and (S3) 135 cm; and two levels of phosphate: (P1) no phosphate, and (P2) 25 kg/ha phosphate. She used three different field locations in which to grow the beans. At each location she prepared six plots such that each combination of row spacing with phosphate could be used. The treatment combinations were randomly assigned to plots. Bean yields in kg/plot were measured for each plot. (Source: Petersen 1994: 86)

  3. A food legume agronomist wanted to study the effect of three production factors--variety, phosphorus fertilization, and weed control--on the yield of chick-peas.  The following factor levels were selected: Variety at two levels: V1 and V2; Phosphorus at three levels: P1 = none, P2 = 25 kg/ha, P3 = 50 kg/ha; Weed control at two levels: W1 = none, W2 = herbicide. He believed these factors might not act independently so he constructed the following treatment levels: (1) V1P1W1, (2) V1P2W1, (3) V1P3W1, (4) V1P1W2, (5) V1P2W2, (6) V1P3W2, (7) V2P1W1, (8) V2P2W1, (9) V2P3W1, (10) V2P1W2, (11) V2P2W2, (12) V2P3W2. The area in which the trial was to be run included three soil types. He located 12 plots on each of the soil types and assigned the treatment combinations at random to plots within types. The yields of dry chick-peas in kg/plot was recorded for each plot. (Source: Petersen 1994: 96)

  4. Four cultivars, A, B, C, and D, of dwarf beans were selected for a growth study at a specific site. Thirty-two (32) plots were prepared at the site and the cultivars were randomly assigned to plots such that each cultivar appeared on eight plots. The leaf areas in square-centimeters were recorded for each plot three days after germination. (Source: Pearce et al. 1988: 23)

  5. An experimenter had three promising new varieties of maize, referred to here as X, Y, and Z. He wished to compare them with his existing recommended variety, referred to as R. The following table shows the field layout of the plots along with the yields. Yields were measured in cavans per hectare (cavan is a measure of volume in the Philippines). (Source: Pearce et al. 1988: 25)
    6. Y (32.8)
    Z (24.2)
    R (28.5)
    X (26.9)
    R (29.5)
    X (23.7)
    Z (28.0)
    Y (25.8)
    X (33.4)
    R (14.2)
    Y (33.3)
    Z (23.6)
    Z (31.3)
    Y (25.8)
    X (33.1)
    R (13.2)

  6. Another experimenter wished to compare the yields of six strains (A-F) of maize. The following field plan was used with yields measured in bushels per acre. (Source: Pearce et al. 1988: 25-26)
    7. B (20.3)
    C (14.9)
    A (13.7)
    D (14.7)
    E (16.7)
    F (18.6)
    E (18.2)
    A (15.3)
    C (10.1)
    F (15.3)
    B (10.8)
    D (11.3)
    A (12.2)
    E (16.7)
    F (12.2)
    C (11.2)
    D (11.4)
    B (10.5)
    D (18.9)
    F (17.9)
    E (16.8)
    B (16.0)
    A (14.3)
    C (16.4)
    F (17.7)
    B (20.1)
    D (17.0)
    A (15.3)
    C (15.5)
    E (17.0)
    C (18.7)
    D (17.9)
    B (19.9)
    E (17.7)
    F (18.4)
    A (12.1)

  7. Five varieties of spring wheat were sown on plots located within four blocks controlling for variation across a field. The soil of each plot was treated with three different levels of nitrogen randomly allocated to equal areas within each plot. The field layout and yields in tons per hectare are given in the table below. (Source: Pearce et al. 1988: 164-165).
    8. Block
    I
    V2
    N1 N3 N2
    4.6 5.5 5.3
    V5
    N2 N3 N1
    5.0 5.4 4.7
    V1
    N1 N2 N3
    5.5 6.1 6.4
    V4
    N1 N3 N2
    5.0 6.0 5.7
    V3
    N2 N1 N3
    5.5 4.9 5.8
    Block
    II
    V1
    N3 N1 N2
    5.8 5.0 5.5
    V3
    N1 N3 N2
    4.9 5.5 5.4
    V2
    N3 N2 N1
    5.4 5.0 4.7
    V5
    N2 N3 N1
    4.6 5.0 4.2
    V4
    N2 N1 N3
    6.2 5.7 6.5
    Block
    III
    V5
    N2 N3 N1
    4.8 5.0 4.6
    V1
    N2 N3 N1
    5.4 5.9 5.0
    V3
    N3 N1 N2
    5.5 4.8 4.7
    V2
    N1 N3 N2
    5.0 5.8 5.1
    V4
    N1 N3 N2
    5.3 6.7 5.8
    Block
    IV
    V2
    N3 N1 N2
    5.9 5.0 5.6
    V3
    N3 N2 N1
    4.8 4.6 4.0
    V4
    N2 N1 N3
    5.1 4.7 5.4
    V1
    N1 N2 N3
    5.2 5.5 5.8
    V5
    N3 N2 N1
    5.2 4.8 4.4

  8. An experiment was undertaken to study the yields of hay grass under five different cutting regimes (A-E) and four different grass mixtures (W, X, Y, and Z). Cutting regimes were assigned to plots according to the diagram below and then grass mixtures were randomly assigned to quarter plots of each full plot (again, see the diagram, although the randomized order of these plots is not shown). With treatments A-C crops of hay were taken on 24th June, 15th June, and 31st October. With A, crops had been taken on three previous occasions, with B on two and with C on one. With treatments D and E, crops were taken with relation to the emergence of ears; with D at emergence and 28 and 70 days afterwards; and with E at 28 and 84 days after emergence. The data are the dry weight matter of hay removed from each quarter plot. (Source: Pearce et al. 1988: 165-166)
    9. C
    W 648
    X 532
    Y 323
    Z 434
    B
    W 453
    X 463
    Y 294
    Z 309
    E
    W 1032
    X 933
    Y 1215
    Z 827
    D
    W 562
    X 540
    Y 407
    Z 511
    A
    W 452
    X 427
    Y 439
    Z 449
    D
    W 392
    X 528
    Y 493
    Z 299
    E
    W 781
    X 759
    Y 588
    Z 890
    A
    W 739
    X 826
    Y 632
    Z 550
    C
    W 630
    X 568
    Y 567
    Z 523
    B
    W 624
    X 490
    Y 618
    Z 508
    A
    W 489
    X 620
    Y 400
    Z 476
    C
    W 499
    X 551
    Y 428
    Z 294
    D
    W 529
    X 673
    Y 422
    Z 676
    B
    W 456
    X 366
    Y 554
    Z 510
    E
    W 1204
    X 958
    Y 967
    Z 950
    B
    W 619
    X 976
    Y 509
    Z 457
    A
    W 378
    X 436
    Y 354
    Z 336
    C
    W 759
    X 734
    Y 457
    Z 602
    E
    W 826
    X 1104
    Y 866
    Z 1380
    D
    W 432
    X 677
    Y 383
    Z 449
    E
    W 1036
    X 765
    Y 474
    Z 704
    D
    W 289
    X 511
    Y 339
    Z 256
    B
    W 626
    X 445
    Y 601
    Z 466
    A
    W 911
    X 918
    Y 704
    Z 551
    C
    W 596
    X 1020
    Y 438
    Z 632

References