# Block design or completely randomized?

I have 5 conditions in my experiment: A,B,C,D,E and my sample size is 200 subjects. I want to have 40 subjects per condition, so I assign a condition to a person randomly without replacement. E.g. Person 1 gets condition B, so I still need 40 people for conditions A,C,D,E and 39 for condition B. Now person 2 has a slightly smaller chance to get condition B again. I don't have a prespecified list of say EABCDDEBAC... etc., where I cycle through the five options It's a random allocation for every person until the 40 people per condition fill up. So I am confused whether this is a block design or completely randomized design?

If I understand your description, this is a completely randomized design. Of course, if you're sampling without replacement, the number of options on each draw dwindles as you proceed. Finally, the one remaining subject automatically gets the last slot. But that last subject had an equal chance of being chosen first--or at any other step along the way.

One way to randomize is to put 200 slips of paper in a hat, each with a subject's name. Mix them up. Mix them up again. Then draw slips out of the hat (without peeking) and put them into five stacks for conditions A, B, C, D, and E.

Another way is to use a computer program (R here) randomly to scramble the 200 numbers. You have a numbered list of the 200 subjects from 1 though 200. The subjects with numbers in the first two rows below are assigned to A, the next two rows to B, and so on. Record the seed (or print the output on paper) and save the numbered list, so you can prove (if necessary) that you assigned subjects to conditions at random.

set.seed(2020)
x = sample(1:200)
x
[1] 130  79 123  94  27  14  26  76   1 119 146 141 156  80  77 100 177 120 185  49
[21]  36 195 190 167 188  30  62 200  78  96 164  13 161 159 192  61   2 153  85 180
[41] 158 168 137 139  98  89  73 108 122 194  87  43 134 186 121 166  64  33 172  69
[61] 175 142  32  59 118  83 148 147  86  22 187 115  50  60  67  37 184   8 135 150
[81]  99  12 125  47  45  20 196  41 179  25  66 162  39 132  54 129 157  17  71  44
[101] 149 114  72  91 111  16  65  74  51 133 128   3  38  81 198  35  24 163 160  42
[121] 104 105 117 197   5  68  93  90 127 101  46 124  56 171  28  15 109 138 189 170
[141]  48 144  75 106 155  82  21 136  10   4 169  29 165 178  11  52  23  53 151 110
[161] 140 193  19  18 174  63 199  97 152  88  40 182 145   6 102 176  84  58  95 131
[181] 126 103 113 143  70 107 154 173  55  31 112   9 183 116 191 181  34  57   7  92


In your case, the "treatment" is the condition that you assign to the subjects at random. As Bruce explained, this is simply a randomized assignment of the treatment but not a blocking factor.

A blocking factor is a part of an experimental design where you control a specific part of the experiment, so that it doesn't confound the results. It is specifically meant to avoid having "nuisance" factors add noise or variability in your results.

Say, originally the conditions are not randomly applied on weekdays, but that they follow a specific order so that

• condition all group A on Monday
• condition all group B on Tuesday
• condition all group C on Wednesday
• condition all group D on Thursday
• condition all group E on Friday

Then, day of the week would be an example of a "nuisance" variable. In this case, the day of the week might add noise/interfere with the results (i.e., maybe people are happier Friday vs. Mondays potentially confounding with the Tx effect).

However, if you don't want to study but you don't want it to affect your results, you have to control against it, so you "block" it and now you have to control against. Say by saying

• assign 8 folks in each of groups A,B,C,D,E Monday
• assign 8 folks in each of groups A,B,C,D,E Tuesday
• assign 8 folks in each of groups A,B,C,D,E Wednesday
• assign 8 folks in each of groups A,B,C,D,E Thursday
• assign 8 folks in each of groups A,B,C,D,E Friday

And now, each condition's results are not confounded by day of the week, since they've been assigned across the week days.